Condition and Homology
in Semialgebraic Geometry
vorgelegt von
M. Sc.
Josué T onelli Cuet o
 ORCiD: 0000-0002-2904-1215
von der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promo tions aus schuss :
Vorsitzender: Prof. Dr. John M. Sulliv an
Gutachter: Prof. Dr. Peter Bürgisser
Prof. Dr. Felipe Cucker
C.R. Dr. Pierre Lairez
Tag der wissenschaftlichen Aussprache: 28. November 2019
gefördert durch die Einstein Stiftung Berlin
im Rahmen des Einstein Visiting Fellowships von Felipe Cucker
Berlin 2019

iii
Abstract
The computation of the homology groups of semialgebraic sets (given by Boolean for-
mulas) remains one of the open challenges of computational semialgebraic geometry. De-
spite the search for an algorithm taking singly exponential time only on the number of vari-
ables, as of today, the existing algorithms are symbolic and doubly exponential. In this PhD
thesis, we show how to obtain a numerical algorithm running in single exponential time with
very high probability, which improves the state-of-the-art. To do so, we explain the underly-
ing ideas, methods and techniques from numerical algebraic geometry, numerical complex-
ity and topological data analysis that made this progress possible. We finish with a list of
open problems and questions pointing to a possible future of the numerical computation of
topological invariants.
Additionally, in the appendices, we cover the topic of the expected number of real zeros
of a random fewnomial system and we give an accessible account of the central theme in
Spanish.
Zusammenfassung
Die Berechnung der Homologiegruppen von semialgebraischen Mengen (gegeben durch
boolesche Formeln) bleibt eine der offenen Herausforderungen der algorithmischen semial-
gebraischen Geometrie. Trotz der Suche nach einem Algorithmus mit einfach exponentieller
Laufzeit in der Anzahl der Variablen, sind die nach heutigem Stand bekannten Algorithmen
symbolisch und doppelt exponentiell. In dieser Doktorarbeit zeigen wir, wie man einen nu-
merischen Algorithmus konstruiert, der mit großer Wahrscheinlichkeit einfach exponentiell
ist und somit den Stand der Forschung verbessert. Dazu erklären wir die zugrundliegenden
Ideen, Methoden und Techniken von numerischer algebraischen Geometrie, numerischer
Komplexität und topologischer Datenanalyse, die dieser Fortschrift möglich machten. Wir
enden mit einer Liste offener Probleme und Fragen, die auf eine mögliche Zukunft von Be-
rechnung der topologischen Invarianten weisen.
Außerdem, behandeln wir im Anhange die erwartete Anzahl reeller Nullstellen eines zu-
fälligen Systems polynomialer Gleichungen mit wenigen Termen und geben einen informellen
Überblick über das Hauptthema auf Spanisch.

iv
Laburpena
Multzo semialjebraikoen (formula boolearrek emandakoen) homologia-taldeak kalkula-
tzeak jarraitzen du, oraindik ere, geometria semialjebraiko konputazionalaren erronka han-
dienetako bat izaten. Bilatzen den algoritmoak aldagai kopuruan baino ez du hartzen den-
bora behin esponentziala; hala ere, gaur egun dauden algoritmo guztiak sinbolikoak eta bi
aldiz esponentzialak dira. Doktorego-tesi honetan erakusten dugu nola lor daitekeen debora
behin esponentzialean eta probabilitate handiarekin exekutatzen den zenbakizko algoritmo
bat; hori teknikaren egoeraren hobekuntza da. Horretarako, zenbakizko geometria aljebrai-
koaren, zenbakizko konplexutasunaren eta datu-analisi topologikoaren azpian dauden eta
aurrerapen hori posible egin duten ideia, metodo eta teknikak azaltzen ditugu. Problemen eta
galdera irekien zerrenda batekin bukatzen dugu, zeinek inbariante topologikoen zenbakizko
konputazioaren etorkizun posible bat adierazten baitute.
Gainera, eranskinetan, ausazko sistema oligonomiko baten zero kopurua aztertzen du-
gu, eta tesi honen ikuspegi informala ematen dugu gaztelaniaz.
Resumen
El cálculo de los grupos de homología de conjuntos semialgebraicos (dados por fór-
mulas booleanas) es todavía uno de los mayores desafíos de la geometría semialgebraica
computacional. Aunque se busca un algoritmo que tome a lo sumo tiempo simplemente
exponencial en el número de variables, hasta el día de hoy todos los algoritmos existentes
son simbólicos y doblemente exponenciales. En esta tesis doctoral, mostramos cómo se
puede obtener un algoritmo numérico que tome tiempo simplemente exponencial con al-
ta probabilidad, lo cual es una mejora del estado del arte. Para ello, explicamos las ideas,
métodos y técnicas subyacentes procedentes de la geometría algebraica numérica, de la
complejidad numérica y del análisis topológico de datos que han hecho posible este pro-
greso. Terminamos con una lista de problemas y preguntas abiertas que indican un posible
futuro de la computación numérica de invariantes topológicos.
Además, en los apéndices, estudiamos el número esperado de ceros reales de un
sistema oligonómico aleatorio y damos una visión informal del tema principal de esta tesis
en castellano.

A Евгения,
por todo el apoyo y comprensión durante este duro camino
A Leviathan,
por esperarme casi todas las noches aunque llegara de madrugada

A todas aquellas personas que me han acompañado durante todos estos años en Berlín,
although not exclusively, particularly to:
Nela
Justine
Ana
András
JD
Eddie
Giulia
Marek
Rinat
Shane
Qianheng
Robbie

vii
Acknowledgments, agradecimientos
eta beste esker onak
Agradezco a mis dos directores de tesis Peter Bürgisser and Felipe Cucker for their
constant support, scientific guidance. mathematical discussions and the opportunity to do
my PhD under their supervision. Alde batetik, I give thanks to Peter for his attention to detail
and his Swiss clock-maker’s precision in both mathematics and mathematical writing, eta
beste aldetik, Felipe eskertzen dut por todas sus historias, su estilo de trabajo centrado y
constante y su acogida tanto en Hong Kong como en Tuileries d’Affiac. At last but not least,
I specially give them thanks for passing on their love for complexity, numerical algorithms
and random algebraic geometry on to me.
Figure FP : Peter Bürgisser (right) and Felipe Cucker (left) drawn by Jorge Chan
I must specially thank Alperen Ergür for all the days we have shared in the office in Berlin, his
mentoring during these years, and all our conversations, mathematical or not. I can certainly
claim that without the latter, I would have not developed as much as I did as a mathematician
and a person.

viii Josué Tonelli-Cueto
I also want to take this chance to specially thank the secretary of the group Beate Nießen
for her help with the bureaucracy, which has made my academic life a lot easier than what
it would have been without her help.
Some parts of this thesis employ extensively words and expressions que vienen de
otras lenguas distintas a la inglesa. Because of this, I must thank the people who checked
those. Agradezco a Marta Macho-Stadler por la revision del resumen en castellano, a Julio
García de los Salmones (also known as “Puño de Hierro”) por el resumen en euskera; y a Ana
Barrena Lertxundi und Philipp Reichenbach por el resumen en alemán. Aditionally, I want to
give thanks to Qianheng Cheng for checking my use of Chinese in this writing, to Evgenia
Lagoda for checking my use of Russian; and to Mariane Eguía Sarachaga, Justine Mullon
and Pierre Lairez for useful discussions about how the title of Chapter 3 should be.
Regarding the thesis, I want to thank Matías Bender, Ricardo Grande and Evgenia
Lagoda for useful comments about the introduction (Chapter 0 ) of this thesis, Aliciacia Dick-
enstein for answering my questions regarding her latest work with Bihan and Forsgård [59] ,
Teresa Krick for pointing many typos in the read version, Pierre Lairez for providing me with
a digital copy of the master’s thesis of his student Jiadong Han [213] , Marta Macho-Stadler
for extensive comments on Appendix M , and Frank Sottile for giving me access to the doc-
uments attached to the letter of Kushnirenko [268] . Also, I am thankful to Pierre Lairez and
to John M. Sullivan for accepting being part of my doctoral committee.
Regarding my academic stays, I want to thank Nicolai Vorobjov for kindly hosting me in
Bath for a research visit, and Elias Tsigaridas for his invitation to Paris. Regarding the personal
side of these stays, estoy enormemente agradecido por la acogida y el cariño ofrecido por
Antonia, Alfonso, Matías, Tomás y Tatiana.
In these years, many people has accompanied and supported me, either directly or
indirectly, either by direct interaction or action at a distance. Even if I don’t mention them by
name, I am thankful to all of them.
In Berlin, I want to particularly thank the following people: Nela, Justine, Eddie, JD, Rinat,
Marek, Shane, Matthias, Robbie, Max, Flavie, Tiho, Mia and Lorena, for all the nights out and
all the shared moments; Ana, Dani y Zara, por todos los momentos compartidos durante y
después del Soulkino; les compañeres del Círculo Podemos Berlin, especialmente Concha,
Raquel, Lluis, Cristina, Pedro, Mariángeles, y de nuevo, Ana y Zara; András, Giulia, Qianheng,
Héctor, Marco, Yuya, Levent, Duca and Jorge, for the shared experience in and out of the
BMS; Alex and Sophia, for all the movie and aftermovie moments; Qiao, Brent, and again,
Héctor and Yuya, for our joint venture as BMS Student Representatives and organizers of the
6th BMS Student Conference; Tommaso, Georg, Irem, Asilya, Konstantin, Barbara, Stefan,
Riccardo, Simona, Johanna, Michael, Gregor, and again, András, Giulia, Qianheng, Marco,
Duca and Levent, for the collective effort in the organization of the “ ®
w h α t i Ó . . . ?” Seminar;
and Matías, Philipp, Paul, Sascha, Mario and Philipp, for the daily life in the working group.
Fuera de Berlín, quiero agaradecer particularmente a las siguientes personas: Aritzeder,
por seguir siendo mi amigo tras tanto tiempo estando siempre ahí; Marta, Ricardo, Iratxe,
Tere, Mikel, Adrián, Maider, Leire, Ion y Miren, por mantener la amistad y el contacto en la
distancia; Oihane, Julene, Andrea, y de nuevo, Iratxe y Maider, por mantener el contacto y
encontrar tiempo para reunirnos cada año para rememorar viejos tiempos; mis profesores

Condition and Homology in Semialgebraic Geometry ix
de la Licenciatura en Matemáticas, en particular, Charo, Luis, Raúl, Ion, Gustavo, Josu,
Virginia, Lourdes, Pedro, Eugenio, y de nuevo, Marta, por lo que les debo en mi formación
matemática; Jose y Raimundo, por su aprecio desde que soy niño; mi familia, Juan, Isabel,
Pablo, Elena, Ana Lara, Paco, Anita, Ramonín y Ramón, por estar ahí; y, especialmente, mis
padres, por todo.
And concluding in the obvious way, very special thanks to Евгения and Leviathan.

xi
The completion of this PhD Thesis and the Phase II of the Berlin Mathematical School
(BMS) was possible thanks to the doctoral fellowship awarded to me by the Berlin Mathemat-
ical School within the framework of Felipe Cucker’s Einstein Visiting Fellowship “Complexity
and accuracy of numerical algorithms in algebra and geometry” and which was funded by
the Einstein Foundation Berlin .
Der Abschluss dieser Doktorarbeit und der Phase II an der Berlin Mathematical School
(BMS) wurde durch ein Stipendium ermöglicht, das mir die Berlin Mathematical School im
Rahmen des Einstein Visiting Fellowships ” Complexity and accuracy of numerical algorithms
in algebra and geometry “ von Felipe Cucker vergab und welches durch die Einstein Stiftung
Berlin gefördert wurde.

xiii
Assumptions and conventions
Personal Pr onoun Pr onouncement
Following Spivak [Q16 ; Personal Pronoun Pronouncement ] , we will use a genderless
pronoun, now known today as the Spivak pronoun , to avoid the gender specification when-
ever we refer to a person of undetermined gender (like a random mathematician or the
reader). In this way, we will use “E” instead of “he or she” or “they”, “Em” instead of “him or
her” or “them”, and “Eir” instead of “his or her” or “their”.
Conventions on translations and transliterations
T ranslations Whenever we have consulted a translated work, we cite the translation
together with a note indicating which work is a translation of instead of adding a reference
to the original work also. See [413] for an example.
T ransliterations to the Latin alphabet The general convention has been to use the
transliteration preferred by the author if possible, and the most accepted one otherwise.
This allows us to be consistent and to avoid referring to the same author by several names.
However, this means that the used spelling of the romanized name might differ from that of
a particular referred reference, which will be probably the case for names that have changed
their transliteration over time (such as names with Cyrillic spelling). If the cited reference is
written in the author’s mother tongue, we additionally indicate in parenthesis the spelling of
the author’s name in the original alphabet (as it can be seen in the reference [414] ).
Assumptions on the r eader and mathematical conventions
This thesis, as any other mathematics text in history and in the world, will assume cer-
tain knowledge on the part of the reader. With the exception of the last appendix, which
requires Spanish knowledge, but no mathematical knowledge; the thesis will assume on
the reader the ability to read and understand English 1 , some mathematical knowledge and
certain mathematical maturity. The latter should be interpreted as having an ability to follow
and understand mathematical ideas and proofs and experience reading mathematics at the
graduate level at least. The mathematical knowledge needed and some conventions that we
will use are explained below.
Algebraic geometry We will not assume any knowledge in algebraic geometry be-
yond the basic notions such as zero sets and polynomials. A knowledge in real and semial-
gebraic geometry will be useful to understand the motivation of certain questions from the
1 If the reader has arrived to this point, it means that probably E satisfies this requirement or that E likes to
stare at sequences of characters that are incomprehensible for Em.

xiv Josué Tonelli-Cueto
algebraic geometric perspective, but any necessary prerequisites will be introduced, par-
ticularly the notions of semialgebraic set and condition number of real projective algebraic
sets.
Algebraic topology We will assume that the reader is familiar with the basics of al-
gebraic topology: homotopies, continuous retractions, homotopy equivalences, singular ho-
mology and the Mayer-Vietoris theorem in homology. We will not require any knowledge on
homotopy groups, beyond their definition and the fact that they are preserved under homo-
topy equivalences. The reader can find any unknown notion in the standard references [216]
and [346] .
Complexity theory We will assume some familiarity with complexity theory, in the
sense that we assume that the reader is familiar with how the time complexity of an algorithm
is estimated in general. The only point of the thesis were a serious knowledge of complexity
theory is needed is in the subsection 0 §2 -3 where we give the computer scientific motivation
of the problem that this thesis discusses.
Dif fer ential and Riemannian geometry We will assume that the reader is comfort-
able and familiar with the standard concepts of differential and Riemannian geometry that
are covered in a usual graduate course in mathematics. The reader can find any unknown
notion in the standard references [275] and [381] .
We will be working mainly on the sphere Ó n and Ò n +1 . To be clear, for any smooth
map f : Ò n +1 → Ò q , including polynomials, we will denote by D x f the tangent map
T x Ó n → Ò q of f , as map on the sphere Ó n , at x ∈ Ó n and by D x f the tangent map
T x Ò n +1 = Ò n +1 → Ò q of f , as a map on Ò n +1 , at x ∈ Ò n +1 . When f : M → N is
a smooth map between smooth manifolds M and N such that either M or N is not an
Euclidean space, we will denote by D x f the tangent map T x M → T f ( x ) N of f at x ∈ M .
Linear algebra We will assume that the reader is familiar with the Singular Value De-
composition (SVD), singular values and orthogonal and unitary transformations. The reader
can find any unknown notion in [392] .
Pr obability theory We assume that the reader is familiar with the basic notions of
probability theory. By this, we mean that the reader must know the definition and interpreta-
tion of probability, random variables and vectors and expectations in the continuous setting.
The reader can find any unknown notion in [164 ; Ch. 1 ] .

xv
Notations
[ m ] := { 1 , . . . , m }
# A cardinal/size of A
[ m ] ≤ l := { A ⊆ [ m ] | # A ≤ l }
sgn sign map p. 68
A | B : = { A ∩ B | A ∈ A } , for A collection of sets
≼ boundary order on { − 1 , 0 , +1 } q ( 2.10 )
Parameters
q number of polynomials
d := ( d 1 , . . . , d q )
D := max i ∈ [ q ] d i
N i := ( n + d i
d i )
N := ∑ q
i =1 ( n + d i
d i ) = ∑ q
i =1 N i
∆ d := diag ( √ d ) ( 1.8 )
Polynomials
P d [ q ] ( 0.3 )
H d [ q ] d -homogeneous polynomial q -tuples in X 0 , . . . , X n ( 1.1 )
∥·∥ W Weyl norm ( 1.2 )
⟨ · , · ⟩ W Weyl inner product ( 1.3 )
ev i
x , dev i
x , v ( 1.4 )
R x ( H d [ q ]) , L x ( H d [ q ]) , C x ( H d [ q ]) Corollary 1 §1 3
R x , R 0
x , R 1
x Proposition 1 §1 6
f u := f ( u X ) ( 1.5 )
f Ó ( 1.18 )
p h homogenization map ( 1.28 )
H ( p ) ( 1.30 )
H ∞
d [ q ] p. 54
p h ( 1.33 )
b
f ( 5.13 )
c
+ f ( 5.13 )

xvi Josué Tonelli-Cueto
Boolean formulas
Φ , Ψ , . . . Boolean formulas (generally)
ϕ , ψ , . . . purely conjunctive formulas (generally)
Φ X ( S 1 , . . . , S a ) value in X of Φ at ( S 1 , . . . , S a ) ⊆ X a p. 4
Φ h p. 53
H (Φ) ( 1.31 )
NF ( ϕ ) normal form Proposition 2 §1 2 .1
DNF (Φ) disjunctive normal form Proposition 2 §1 2 .2
sgn ( ϕ ) sign vector of ϕ p. 71
s DNF (Φ) strict disjunctive normal form Lemma 2 §2 3
Φ ГВ δ , ε ( 4.11 )
Φ ГВ δ , ε ( 4.12 )
Zer o and semialgebraic sets
Z Ó ( f ) zero set of f in the sphere Ó n
W ( p , Φ) semialgebraic set described by ( p , Φ) ( 0.2 )
S ( f , Φ) spherical semialgebraic set described by ( f , Φ) ( 1.22 )
Z Ó
r ( f ) algebraic neighborhood of Z Ó ( f ) ( 2.4 )
S ( f , t , Φ) spherical semialgebraic set described by ( f , t , Φ) ( 2.5 )
S r ( f , t , Φ) algebraic neighborhood of S ( f , t , Φ) ( 2.6 )
ГВ δ , ε ( f , Φ) Gabrielov-Vorobjov ( δ , ε ) -block Definition 2 §4 1
ГВ δ , ε ( f , Φ) Gabrielov-Vorobjov ( δ , ε ) -approximation Definition 2 §4 1
X ( f , t , Φ , G ) approximating cloud of G -points for ( f , t ) ( 4.1 )
X ∝
i , j ( f , t , G ) Definition 4 §1 2
Z Ã ( f ) complex zero set of f
Z ( f ) zero set of f
Metric notions
dist Euclidean distance
B ( x , r ) Euclidean ball with center x and radius r
B ( x , r ) closed Euclidean ball with center x and radius r
U ( X , r ) (Euclidean) r -neighborhood ( 3.2 )
dist Ó geodesic distance on Ó n ( 1.9 )
B Ó ( x , r ) ball with center x and radius r with respect to dist Ó
B Ó ( x , r ) closed ball with center x and radius r with respect to dist Ó
U Ó ( X , r ) spherical r -neighborhood ( 2.3 )
dist W Distance with respect the Weyl norm
B W ( f , r ) ball with center f and radius r with respect to dist W
dist H Hausdorff distance ( 3.1 )
dist X distance to X function

Condition and Homology in Semialgebraic Geometry xvii
Dif fer ential geometry
Ò n +1 ( n + 1) -dimensional Euclidean space
Ó n n -dimensional sphere
Ó n
+ upper half n -dimensional sphere
Ó n
0 ( n − 1) -dimensional sphere given by Z Ó ( X 0 )
T x M tangent space of M at x
N x M normal cone of M at x (in its ambient space)
D x f tangent map T x M → T f ( x ) N
tangent map T x Ó n → T f ( x ) N , if f polynomial tuple
D x f tangent map Ò m → Ò m ′ at x
Ю ( 1.29 )
Ю ( 4.20 )
+ f gradient vector of f
Linear algebra
1 vector of ones
É identity matrix
∥ · ∥ F Frobenius norm ( 1.6 )
⟨ · , · ⟩ F Frobenius inner product ( 1.6 )
∥ · ∥ operator norm ( 1.7 )
σ i ( A ) i th singular value of A
A ∗ (conjugate) transpose of A
A † pseudoinverse of A ( 1.12 )
SNF ( A ) Smith Normal Form of A ( 3.23 )
O ( n ) orthogonal group of order n
U ( n ) unitary group of order n
h x ( 4.23 )
Condition numbers and r elatives
κ ( f , x ) local condition number of f at x ( 1.10 )
κ ( f ) global condition number of f ( 1.11 )
µ ( f , x ) ( 1.15 )
γ ( f , x ) Smale’s gamma ( 1.16 )
γ ( f , x ) Smale’s projective gamma ( 1.17 )
κ ( f , x ) local intersection condition number of f at x ( 1.23 )
κ ( f ) global intersection condition number of f ( 1.24 )
κ aff ( p ) global affine intersection condition number of p
κ ∞
aff ( p , x ) ( 1.36 )
κ ∞
aff ( p ) ( 1.36 )
Щ ( t ) separation of t ( 2.7 )
κ aff ( f , x ) local affine condition number of f at x Definition 5 §2 2

xviii Josué Tonelli-Cueto
Discriminat sets
Σ d [ q ] x local discriminant set at x ( 1.13 )
Σ d [ q ] global discriminant set ( 1.14 )
Σ d [ q ] x ( 1.26 )
Σ d [ q ] ( 1.26 )
Σ L
d [ q ] x ( 1.27 )
Σ L
d [ q ] ( 1.27 )
Σ aff
d [ q ] ( 1.34 )
Σ aff
d [ q ] + ( 1.35 )
Σ aff
d [ q ] 0 ( 1.35 )
Reach, ˇ
C ech, V ietoris-Rips and r elatives
π X nearest point retraction of X ( 3.3 )
∆ X medial axis of X ( 3.4 )
+ X ( 3.5 )
τ ( X , x ) local reach of X at x ( 3.6 )
τ ( X ) reach of X ( 3.7 )
τ ( X , x ; u ) local reach along u of X at x ( 3.8 )
C
 ε ( X ) C
 ech complex of X of radious ε ( 3.24 )
c  π ( 3.25 )
V R ε ( X ) Vietoris-Rips complex of X of radious ε ( 3.26 )
ϑ m ( 3.27 )
Ç V R
ε ( X ) Vietoris-Rips graph of X of radious ε ( 3.28 )
Algebraic topology
H • (singular) homology
H k k th (singular) homology group ( 0.1 )
β k k th Betti number ( 0.1 )
⊤ k k th vector of torsion coefficients ( 0.1 )
s k number of entries in ⊤ k ( 0.1 )
π k k th homotogy group
N ( C ) nerve of C ( 3.12 )
∆ X free simplex with vertex set X ( 3.13 )
[ S ] realization of a simplicial complex S ( 3.14 )
dim σ dimension of face σ ( 3.15 )
S k set of k -kaces of S ( 3.16 )
C ∆
k ( S ) set of simplicial k -chains of S ( 3.17 )
∂ ∆
k k th boundary operator ( 3.18 , 3.19 )
B ∆
k ( S ) set of simplicial k -boundaries of S ( 3.20 )
Z ∆
k ( S ) set of simplicial k -cycles of S ( 3.21 )
H ∆
k ( S ) k th simplicial homology group of S ( 3.22 )
β p
k k th mod p Betti number ( 4.29 )

Condition and Homology in Semialgebraic Geometry xix
Pr obability theory
Ð probability
Å expectation
Å x ∈ K expectation over the uniform distribution on K
x , y random vector
A random matrix
f , g , p random polynomial tuples
N ( x , σ ) normal distribution centered at x with standard deviation σ
U ( Ó N − 1 ) uniform distribution on the sphere Ó N − 1
χ 2
m χ 2 -distribution with m degrees of freedom
L x concentration function of x ∈ Ò k ( 5.3 )
F k , l Fisher-Snedecor distribution with k and l degrees of freedom
Dif fer ential tools
N f
x Newton vector field of f ( 2.1 )
N f , t , ϕ
x discontinuous Newton vector field of ( f , t , ϕ ) ( 2.8 )
Л f , λ ( f , λ ) -lartition Definition 2 §2 4
л J strata of Л f , λ associated to J Definition 2 §2 4
П f , λ ( f , λ ) -partition Definition 2 §2 5
п I , σ strata of П f , λ associated to ( I , σ ) Definition 2 §2 5
Special functions
Γ Euler’s Gamma function
Grids
G й uniform grid of order й ( 4.19 )
O й ( я ) random й-grid with failure probability я − 1 p. 143
R й recursive й-grid with seeds R 0 and N ( 4.24 )
Subdivision methods
□ [ X ] boxes ∏ [ a i , b i ] included in X
□ [ F ] interval approximation of F ( 5.4 )
m ( J ) midpoint of J ∈ □ [ X ]
w ( J ) maximum width of J ∈ □ [ X ]
C f ( J ) ( 5.5 )
C ′
f ( J ) ( 5.14 )
Err or analysis
F, F e 0 , e 1
b , t floating-point number system ( 4.30 )
u , u b , t round-off unit Definition 4 §3 1
fl, fl b , t rounding map ( 4.31 )
f
op approximate version of op ( 4.32 )
fl a p. 163
з ( k ) ( 4.33 )
1 + J · K p. 163

xx Josué Tonelli-Cueto
Other
ж ( θ ) ( 4.13 )
э ( θ ) ( 4.14 )
д ( θ ) ( 4.15 )

xxi
Contents
Abstract / Zusammenfassung / Laburpena / Resumen . . . . . . . . . . . . . . . iii
Acknowledgments, agradecimientos eta beste esker onak . . . . . . . . . . . . . vii
Assumptions and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
N o t a t i o n s ...................................... x v
0 Intr oduction 1
0 §1 Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0 §2 Motivation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0 §3 State-of-the-art and contributions to the problem . . . . . . . . . . . . . . . 16
AI A n a l y t i c a l i n d e x ................................ 2 2
1 Condition numbers for the homology of semialgebraic sets 25
1 §1 Homogeneous polynomials and Weyl norm . . . . . . . . . . . . . . . . . . 26
1 §2 κ : a condition number for spherical algebraic sets . . . . . . . . . . . . . . 33
1 §3 κ : a condition number for spherical semialgebraic sets . . . . . . . . . . . . 49
1 §4 κ aff : a condition number for affine semialgebraic sets . . . . . . . . . . . . . 53
2 Dif fer ential semialgebraic geometry with condition-based inequalities 59
2 §1 A converse to the Exclusion Lemma . . . . . . . . . . . . . . . . . . . . . 59
2 §2 Mather-Thom theory and some Whitney stratifications . . . . . . . . . . . . 66
2 §3 D u r f e e ’ s t h e o r e m ............................... 8 1
2 §4 Gabrielov-Vorobjov approximation theorem . . . . . . . . . . . . . . . . . . 86
3 Computing the homology of a set par pointillage 99
3 §1 Approximation of sets by clouds of points . . . . . . . . . . . . . . . . . . 100
3 §2 Reach: Explicit lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . 110
3 §3 Homology of a cloud of points: the nerve theorem . . . . . . . . . . . . . . 116
4 Numerical algorithms for the homology of semialgebraic sets 131
4 §1 Simplicial approximation of semialgebraic sets . . . . . . . . . . . . . . . . 132
4 §2 Algorithms and their complexity . . . . . . . . . . . . . . . . . . . . . . . . 140
4 §3 Parallelization and numerical stability . . . . . . . . . . . . . . . . . . . . . 154
5 A look into the futur e 171
5 §1 Beyond normal distributions: robust tail bounds . . . . . . . . . . . . . . . 171
5 §2 1st adaptive case: Plantinga-Vegter algorithm . . . . . . . . . . . . . . . . 179
5 §3 2nd adaptive case: Han’s covering algorithm . . . . . . . . . . . . . . . . . 190
5 §4 A пятилетка for the future . . . . . . . . . . . . . . . . . . . . . . . . . . 198
F Real zer os of random fewnomials 205
F §1 A history of real fewnomial theory . . . . . . . . . . . . . . . . . . . . . . . 206
F §2 Random real algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 211

xxii Josué Tonelli-Cueto
F §3 Probabilistic Kushnirenko Hypothesis III . . . . . . . . . . . . . . . . . . . . 215
F §4 A random real fewnomial theory? . . . . . . . . . . . . . . . . . . . . . . . 222
M ¿Cómo computar la “forma” de un conjunto semialgebraico? 227
M §1 Conjuntos semialgebraicos . . . . . . . . . . . . . . . . . . . . . . . . . . 227
M §2 G r u p o s d e h o m o l o g í a ............................. 2 3 1
M §3 C á l c u l o n u m é r i c o ............................... 2 3 2
Index 235
List of algorithms 239
List of figures 241
Bibliography 243
References for used quotes in chapter heads and rhetorics 275
Used auxiliary resources for L
A
T E X 277

1
Die Wahl eines Stils, einer Wirklichkeit, einer Wahrheitsform, Realitäts- und Rationalitätskriterien
eingeschlossen, ist die Wahl von Menschwerk. Sie ist ein sozialer Akt, sie hängt ab von der
historischen Situation, sie ist gelegentlich ein relativ bewußter Vorgang [...], sie ist viel öfter di-
reketes Handeln aufgrund starker Intuitionen. »Objektiv« ist sie nur in dem durch die historische
Situation vorgegebenen Sinn: auch Objektivität ist ein Stilmerkmal [...].
Paul Feyerabend, Wissenschaft als Kunst
0
Intr oduction
This dissertation presents the numerical approach to “one of the most fundamental open
questions in algorithmic semi-algebraic geometry” [25 ; §5 ] : is there a poly ( q , D ) poly ( n ) -time
algorithm for computing the homology groups (or Betti numbers) of a semialgebraic set in
Ò n described by a Boolean formula involving q real polynomials of degree at most D? Our
presentation will be a synthesis of the currently existing results: the author’s PhD work in [91,
92, 136] , with Peter Bürgisser, Felipe Cucker and Alperen Ali Ergür; and the immediately
preceding work in [142, 88] , by Peter Bürgisser, Felipe Cucker, Teresa Krick, Pierre Lairez
and Michael Shub.
Additional PhD work in [93] , with Peter Bürgisser and Alperen Ali Ergür, regarding the
number of real zeros of a random fewnomial system can be found in the Appendix F , since
it does not follow the main thought stream. Also, in Appendix M , an accessible account of
the central theme is given in Spanish 1 .
The present introduction will answer in the following three sections the next three fun-
damental questions:
1. What is precisely the problem we are trying to solve?
2. Why do we care about this problem?
3. What did others do towards the solution of this problem? And what have we done?
After answering these, the analytical index at the end intends to give an overview of the
content and structure of the present thesis.
1 The reason for giving it in Spanish and not in English (or German) is that the target audience (relatives and
friends back in Spain) is Spanish-speaking, but not always English-speaking.

2 Josué Tonelli-Cueto 0 §1
0 §1 Description of the problem
To describe the problem at hand, we review the terms appearing in the question above
and only then, after the shades of meaning have been clarified, state the question with all its
details.
0 §1 -1 Concepts and objects of the problem
The question above mentions three fundamental objects in it: algorithm, homology
groups (and Betti numbers), and semialgebraic sets. These concepts are clear to any expert
in computational semialgebraic geometry, but they may be unclear to a random mathemati-
cian. E might be unfamiliar with some of these notions (as it usually happens with semialge-
braic sets) or might give them a meaning that was not intended (as it can happen with such
a polysemous concept as homology). 2
Algorithm
The notion of algorithm can have many definitions. However, by the Church-Turing the-
sis [12 ; Ch. 1 ] , all of the ones reflecting computation in the real world turn out to be equivalent.
Moreover, strong Church-Turing thesis [12 ; Ch. 1 ] says that all of them have equivalent no-
tions of run-time and computing space, in the sense that the asymptotic complexity classes
of complexity (polynomial, singly exponential, etc.) are the same for all models of computa-
tion. The interested reader can consult the details about the formal models of computation
and their equivalences in any of the standard references, e.g., [12, 308] .
Taking advantage of living in the era of computers, our model of computation will be
pseudo-code. In the usual setting, we would measure the run-time in terms of the number of
bit-operations that the algorithm described by the program in pseudo-code executes. How-
ever, in computational semialgebraic geometry, it is more clarifying to consider an algebraic
model of computation in which algebraic operations and comparisons between real num-
bers can be done at unit cost. Because of this, we will assume that the program executed
by our pseudo-code can perform arithmetic operations and comparisons with real numbers
exactly, we will allow as inputs any real number and we will measure the run-time by the
number of arithmetic operations and comparisons. This could be formalized using BSS ma-
chines [68 ; Ch. 4 ] , but this level of formality will not be required by us as we don’t intend to
prove complexity lower bounds.
The adoption of an algebraic model of computation means stepping out of a realistic
model of computation. To be able to translate our algebraic pseudo-code into a real al-
gorithm, we need to accompany our complexity analysis with a bound on the size of the
approximations/representations of the real numbers that our programs work with. This is
done in two ways:
1. We restrict the real numbers input-output to an efficiently computable class of real
numbers . By which we mean a set of real numbers represented by bit-strings such that
arithmetic operations and comparisons can be done in polynomial-time in terms of the
bit-representations. The paradigmatic example of such a model are rational numbers
2 The claims between parentheses in the paragraph are based on the author’s experience.

0 §1 Condition and Homology in Semialgebraic Geometry 3
with the usual ways we operate and compare them, but there are more sophisticated
versions such as Thom’s encoding of real algebraic numbers [131] . In this restricted
framework, the algorithm becomes efficient (in the bit sense) if one can bound the size
of the representations appearing during the execution of the algorithm.
2. We allow the algorithm to use efficiently computable approximations, by which we
mean a set of real numbers represented by bit-strings such that approximations to
arithmetic operations and comparisons can be done in polynomial-time in terms of the
bit-representation. The paradigmatic examples of this approach are fixed-point and
floating-point arithmetic. In this approximate framework, the algorithm becomes effec-
tive if one can bound the precision needed during the execution in order to guarantee
correct approximation of the output.
In our numerical approach, we do the latter (see Chapter 4 ). Therefore we will be able to
make effective the algorithms that we produce.
Homology gr oups and Betti numbers
By homology, we don’t mean any fancy homological theory from algebraic geometry,
but the singular homology (with integer coefficients) H • ( X )=( H k ( X )) k ∈ Î of a topological
space X [216 ; 2.1 ] . We will refer to this sequence of singular homology groups of X simply
as the homology of X. We note that for simplicial complexes and CW complexes, singular
homology agrees, respectively, with (the more computational) simplicial and cellular homol-
ogy [216 ; Theorems 2.27 and 2.35 ] .
In the cases we will be dealing with, the homology groups will be finitely generated
groups. Therefore, by the classification theorem of finitely generated Abelian groups [273 ;
Ch. III. Theorem 7.7 ] , we have that each homology group H k ( X ) is i somorphic to
Ú β k ( X ) ⊕
s k ( X )
⊕
i =1
Ú
⊤ k , i ( X ) Ú ( 0.1 )
where β k ( X ) and s k ( X ) are natural numbers and the ⊤ k , i ( X ) are positive integers greater
than one such that for all i < s k ( X ) , ⊤ k , i ( X ) divides ⊤ k , i +1 ( X ) . The numbers β k ( X ) , s k ( X )
and the ⊤ k , i ( X ) are uniquely determined by the homology group H k ( X ) . We call β k ( X ) the
k th Betti number of X, and the vector ⊤ k ( X ) := ( ⊤ k , i ( X )) s k ( X )
i =1 the torsion coefficients of X.
We will encode the homology groups through these numbers.
The homology groups H k ( X ) are very robust topological invariants of X, which are not
only invariant under homeomorphisms, but also under homotopy equivalences. Furthermore,
these topological invariants are “easy” to compute for reasonable spaces, when compared
to other topological invariants. However, this easiness of computation comes at a price. Ho-
mology groups are difficult to interpret in direct topological terms, as it is not clear which
topological information they capture. This situation should be compared with that of homo-
topy groups which are difficult to compute (they are still unknown for spheres [416] ), but
which have a direct topological interpretation.
We just briefly recall the interpretation of the first two homology groups and consider
an example. For all topological spaces X, H 0 ( X ) is always free and its rank β 0 ( X ) counts

4 Josué Tonelli-Cueto 0 §1
the number of path-connected components of X [216 ; Proposition 2.7 ] . When X is path-
connected, H 1 ( X ) is isomorphic to the Abelianization of the fundamental group of X, π 1 ( X ) ,
which consists of all the loops based at some point of X up to homotopy equivalence [216 ;
Theorem 2A.1 ] .
A precise description of simplicial homology, how to compute it and examples will be
discussed in Chapter 3 . For more details, we refer the interested reader to any of the usual
references, such as [216, 346] .
Semialgebraic sets
In principle, we could be working over an arbitrary real closed field [70 ; Ch. 1 ] . However,
for the sake of concreteness and numerical algorithms, we will limit ourselves to the real
closed field par excellence : the real numbers Ò from analysis.
A semialgebraic set is a subset of Ò n which can be obtained after performing a finite
number of unions, intersections and complements of sets of the form
{ x ∈ Ò n | p ( x ) = 0 } , { x ∈ Ò n | p ( x ) , 0 } ,
{ x ∈ Ò n | p ( x ) > 0 } and { x ∈ Ò n | p ( x ) ≥ 0 } ,
where p ∈ Ò [ X 1 , . . . , X n ] is a real polynomial. In other words, semialgebraic sets are the sets
which can be described by real polynomials, inequalities and their Boolean combinations.
Example 0 §1 1 . The set of polynomials of the form a X 2 + b X + c with a real zero is a semi-
algebraic set. We can write it as
( { ( a , b , c ) ∈ Ò 3 | a = 0 } ∁ ∩ { ( a , b , c ) ∈ Ò 3 | b 2 − 4 a c ≥ 0 } )
∪ ( { ( a , b , c ) ∈ Ò 3 | a = 0 }∩{ ( a , b , c ) ∈ Ò 3 | b = 0 } ∁ )
∪ ( { ( a , b , c ) ∈ Ò 3 | a = 0 }∩{ ( a , b , c ) ∈ Ò 3 | b = 0 }∩{ ( a , b , c ) ∈ Ò 3 | c = 0 } ) . △
As we cannot feed semialgebraic sets directly to an algorithm, we need to choose a
representation that can be used as an input. We do this by describing semialgebraic sets
with Boolean formulas.
A Boolean formula supported on { α 1 , . . . , α a } is a string Φ constructed recursively by
the following rules:
α k is a Boolean formula ( A )
Φ , Ψ Boolean formulas ⇒ (Φ ∧ Ψ) Boolean formula ( ∧ )
Φ , Ψ Boolean formulas ⇒ (Φ ∨ Ψ) Boolean formula ( ∨ )
Φ Boolean formula ⇒ ( ¬ Φ) Boolean formula ( ¬ )
The atoms of Φ are the α i appearing in it and the size of Φ , size (Φ) , is the number (counted
with repetition) of atoms and operations ( ∧ , ∨ and ¬ ) appearing in Φ . Given sets S 1 , . . . , S a ⊆
X, Φ X ( S 1 , . . . , S a ) is the set obtained by interpreting α i as S i , ∧ as the set-theoretic inter-
section ∩ , ∨ as the set-theoretic union ∪ and ¬ as the set-theoretic complement ∁ in the
ambient set X. When the ambient set X is clear, we omit it.

0 §1 Condition and Homology in Semialgebraic Geometry 5
Remark 0 §1 1 . Taking advantage of the fact that in the operations considered are associative,
we will omit parentheses as long as there is no ambiguity. For example, instead of writing
(((( ¬ a 1 ) ∧ a 2 ) ∨ ( a 1 ∧ ( ¬ a 3 ))) ∨ (( a 1 ∧ a 3 ) ∧ a 4 )) ,
we will write
( ¬ a 1 ∧ a 2 ) ∨ ( a 1 ∧ ¬ a 3 ) ∨ ( a 1 ∧ a 3 ∧ a 4 ) .
Also, since the binary operations are commutative, we can further write ∧ i ∈ I ϕ i to simplify
expressions of the form ϕ i 1 ∧ ϕ 2 ∧ · · · and similarly with ∨ i ∈ I ϕ i . ¶
Given a q -tuple of real polynomials p ∈ Ò [ X 1 , . . . , X n ] q , a Boolean formula over p is a
Boolean formula Φ supported on
{ ( p i = 0) , ( p i , 0) , ( p i > 0) , ( p i ≥ 0) , ( p i < 0) , ( p i ≤ 0) | i ∈ [ q ] } .
Given p ∈ Ò [ X 1 , . . . , X n ] q and a Boolean formula Φ over p , the realization of ( p , Φ) is the
semialgebraic set
W ( p , Φ) := Φ Ò n ( p − 1
i (0) , p − 1
i ( Ò \ 0) , p − 1
i ( Ò > ) , p − 1
i ( Ò ≥ ) , p − 1
i ( Ò < ) , p − 1
i ( Ò ≤ ) | i ∈ [ q ]) ) .
( 0.2 )
A Boolean description of a semialgebraic set is a pair ( p , Φ) where p ∈ Ò [ X 1 , . . . , X n ] q and
Φ is a Boolean formula over p such that W ( p , Φ) = S.
Example 0 §1 2 . In Example 0 §1 1 , the Boolean description suggested is
( ¬ ( a = 0) ∧ ( b 2 − 4 a c = 0)) ∨ (( a = 0) ∧ ¬ ( b = 0)) ∨ (( a = 0) ∧ ( b = 0) ∧ ( c = 0)) . △
Example 0 §1 3 . Let p ∈ Ò [ X 1 , . . . , X n ] q . One can check that
∨
∼ ∈ { >, < } q
q
∧
i =1
( p i ∼ i 0) and
q
∧
i =1
(( p i > 0) ∨ ( p i < 0))
give descriptions for the same set. However, the first formula has size 2 q q − 1 , while the
second one has size 4 q − 1 . This shows that not all descriptions of a semialgebraic set are
equivalent from a computational complexity viewpoint, and that we should be careful with
the assumptions and manipulations of Boolean formulas. △
Remark 0 §1 2 . Boolean formulas can be viewed as formulas or expressions [82 ; 21.5 ] in
the setting of a Boolean algebra. Alternatively, we could have defined a Boolean straight-line
program by changing our description format to that of straight-line programs [82 ; 4.1 ] (also
known as arithmetic circuits when represented as a graph) and everything, including the
proofs and statements in this dissertation, would have carried out in the exact same way.
Although, in general, straight-line programs are more powerful than formulas, it is not
clear that this is the case in the Boolean setting. This is so, because the main example of the
difference, x 2 n , is so due to the ability of straight-line programs to do fast exponentiation.
However, in the Boolean setting, where all binary operations are idempotent, exponentiation
is a useless operation. ¶

6 Josué Tonelli-Cueto 0 §2
0 §1 -2 Statement of the problem
With the above definitions and setting, we can now enunciate precisely the open prob-
lem that concerns this dissertation, which has two versions: one involving only the Betti
numbers (Б) and other involving also the torsion coefficients (В) .
We will write computational problems indicating the input, the output, the complexity
parameters, the desired run-time and the known run-time. The desired run-time indicates
the run-time that the community of experts 3 in computational semialgebraic geometry hopes
for and the known run-time the best existing time-complexity bound.
Let q , n ∈ Î be positive integers, d = ( d 1 , . . . , d q ) ∈ Î q a q -tuple of positive integers,
D := max { d 1 , . . . , d q } and
P d [ q ] := { f ∈ Ò [ X 1 , . . . , X n ] q | for all i ∈ [ n ] , deg f i ≤ d i } , ( 0.3 )
the set of q -tuples f := ( f 1 , . . . , f q ) of real polynomials in the n variables X 1 , . . . , X n such
that f i has at most degree d i .
(Б) : Betti numbers of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output Betti numbers of W ( p , Φ) : β 0 ( W ( p , Φ)) , . . . , β n ( W ( p , Φ))
Complexity parameters s , q , D , n
Desir ed run-time poly ( s , q , D ) poly ( n ) [25]
Known run-time s ( q D ) 2 O ( n ) [128, 419] (cf. [34 ; Ch. 11 ] )
(В) : Homology of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output Betti numbers of W ( p , Φ) : β 0 ( W ( p , Φ)) , . . . , β n ( W ( p , Φ))
Torsion coefficients of W ( p , Φ) : ⊤ 1 ( X ) , . . . , ⊤ n ( X )
Complexity parameters s , q , D , n
Desir ed run-time poly ( s , q , D ) poly ( n )
Known run-time s ( q D ) 2 O ( n ) [128, 419] (cf. [34 ; Ch. 11 ] )
0 §2 Motivation of the problem
There are many ways of motivating a problem. Some of these ways are more appealing
to some people and some to others. Because of this, we don’t present one motivation,
but several of them. We will consider the following four motivations: the applied, because
it may lead to better and faster algorithms in applications; the mathematical, because it will
lead to a better understanding of the class of semialgebraic sets that plays an important
role in many areas of mathematics; the computer scientific, 4 because it plays a central role
in complexity theory; and the historical, 5 because the historical development of real and
3 Or the author, if a citation is not given.
4 Among the motivations, this is the most technical one and it requires some familiarity with complexity theory.
5 Among the motivations, this is the most literary one and also the longest one. The latter is so because this
historical motivations follows very nearly the development.

0 §2 Condition and Homology in Semialgebraic Geometry 7
semialgebraic geometry leads “naturally” to this question. The reader is not supposed to
read all the motivations, but only those that appeal to Eir interests.
0 §2 -1 Applied
In many applications, one is interested in describing the set of possibilities (configura-
tion space) by employing polynomial constraints, i.e., real polynomials and inequalities. This
naturally leads to the appearance of semialgebraic sets in many problems. Without being ex-
haustive in our references, semialgebraic sets play a role in robot motion planning [101, 359,
274] , configurations of molecules [274, 294] , optimization [75, 66, 235, 267] , non-negative
rank [170, 266] , etc.
Example 0 §2 1 (A robot arm) . Consider the following robot arm
o
x
y
z
where the black thick node o is fixed and the grey nodes x , y and z are only fixed to the
position in the bars. The pink bars o y and x z are of fixed lengths 2 and 3 , respectively, and
the red bar o x can change its length from 1 . 5 until 2 . 5 (in the picture is at length 2 ) and rotate
freely between 0 and 90 degrees. The join y is precisely at distance 1 from x and 2 from z .
When the angle and the length of o x vary, the robot arm gets several positions. The
possible configurations of the arm can be encoded as a semialgebraic set. In this case, we
get the following formula
(9 < 4( x 2
1 + x 2
2 ) < 25) ∧ ( x 1 > 0) ∧ ( x 2 > 0)
∧ ( y 2
1 + y 2
2 = 4) ∧ (3 y 1 = 2 x 1 + z 1 ) ∧ (3 y 2 = 2 x 2 + z 2 )
∧ (( x 1 − z 1 ) 2 + ( x 2 − z 2 ) 2 = 9)
where the first line indicate the possible lengths and angles of o x , the second line the length
of o y and the position where y lies on x z and the thirs line the length of x z . △
In many of these applications, once the description as a semialgebraic set has been
obtained, many of the problems reduce to either testing whether the semialgebraic set is
empty (Г) or connected, or counting (Д) or sampling (Е) connected components. Except for
sampling the connected components, all problems reduce to the more general problem of
computing the Betti numbers of a semialgebraic set (Б) .
(Г) : Emptiness of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output 1 if W ( p , Φ) non-empty, 0 otherwise
Complexity parameters s , q , D , n
Desir ed run-time s q n +1 D O ( n ) [25]
Known run-time s q n +1 D O ( n ) [26, 28] (cf. [34 ; Ch. 14 ] )

8 Josué Tonelli-Cueto 0 §2
(Д) : Number of connected components of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output Number of connected components of W ( p , Φ) :
β 0 ( W ( p , Φ))
Complexity parameters s , q , D , n
Desir ed run-time s q n +1 D n · polylog ( n ) [25]
Known run-time s q n +1 D O ( n 2 ) [30] (cf. [34 ; Ch. 16 ] )
(Е) : Connected components of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output x 1 , . . . , x β 0 ( W ( p , Φ)) ∈ W ( p , Φ) s.t.
one and only one x i per connected comp. of W ( p , Φ)
Complexity parameters s , q , D , n
Desir ed run-time s q n +1 D n · polylog ( n ) [25]
Known run-time s q n +1 D O ( n 2 ) [30] (cf. [34 ; Ch. 16 ] )
As of today, direct applications of computing all Betti numbers of a semialgebraic set
are unknown. However, even though direct applications may never appear, the techniques
developed while solving the problems (Б) and (В) might help to provide better algorithms to
any of the problems above or to develop new algorithms for new problems to enter in the
applied world.
0 §2 -2 Mathematical
Semialgebraic sets form a very robust class of sets that remains closed under many
mathematical operations: unions, intersections, complements, projections… Because of this
robustness, the shapes that semialgebraic sets take is vast and is yet to be understood. A
solution to the problems (Б) and (В) would be an advance in its understanding.
However, in current mathematics, the role of semialgebraic sets is not limited to that
of a class of sets that remains to be understood. Semialgebraic sets have a distinguished
position in many areas of mathematics: mathematical logic [386, 362] , where semialgebraic
sets appear in the first-order theory of the reals; real algebraic geometry [47, 70] , where
they appear in any classification problem; complexity theory [69, 68] , where they are central
to real complexity theory; discrete geometry [332] , where they are key to understand the
geometric configurations of a given combinatorial configuration; etc.
Example 0 §2 2 (Realization space of a pyramid) . Consider a square pyramid, with vertices of
the base x 1 , x 2 , x 3 , x 4 and apex y . The realization space should indicate us configurations
of the points in which their convex hull gives the square pyramid combinatorially. In general,
this means ensuring that (1) all points in a facet lie in the same hyperplane and no other point
in that hyperplane, and (2) the points are in convex position. This leads to a semialgebraic
set.
The first part of (1) is translated into equating determinants to zero and the second
part of (1) together with (2) into imposing positivity condition onto determinants, where signs
come from a global orientation. For example, since x 1 , x 2 , x 3 , x 4 lie in the same facet, this

0 §2 Condition and Homology in Semialgebraic Geometry 9
translates into
det ( 1111
x 1 x 2 x 3 x 4 ) = 0 ,
but since y does not lie in this facet this translates, after choosing an orientation of the facet,
det ( 1 1 1 1
x 2 x 3 x 4 y ) > 0 .
Now, we only need to add further positivity conditions as the rest of the faces are triangles.
After choosing an orientation, we get
det ( 1 1 1 1
x i x i +1 x i +2 y ) > 0
where we interpret the subindices mod 4 and i ∈ [4] .
This would give a full description of the point configurations of { x 1 , x 2 , x 3 , x 4 , y } that
gives the pyramid with a square base. However, the realization space is the set of those con-
figurations up to natural transformations, which are usually either projective transformations,
affine transformations or isometries. In order to do this, we just fix as many parameters as
we can using the considered action.
For example, in our case for the affine transformations, we can just assume that x 2 = 0 ,
x 3 = e 1 , x 4 = e 2 and y = e 3 . Doing this, we can see that x = ( a b 0 ) ∗ and that the
affine realization space of the pyramid is described by
( − a > 0) ∧ ( b > 0) ∧ (1 − a − b > 0) .
Although the realization space in this example is simple, they can be arbitrarily complicated
and have any sequence of Betti numbers a simplicial complex can have [332] . △
In this way, solving (Б) and (В) (both in theory and in practice) provides a computational
approach to problems in which the homology of semialgebraic sets plays a fundamental
role. An example of this, in a variant of (Б) for curves, can be appreciated in [241] to re-
study explicitly Gudkov’s classical solution to Hilbert’s sixteenth problem for curves of degree
six [208, 209, 210, 207] . We will come back to this in Chapter 5 .
0 §2 -3 Computer scientific
A counting problem refers to a computational problem involving a function that can be
interpreted as the counting function for some object. The most famous class of counting
problems is # P , but, in (computational) complexity theory, one is also interested in counting
problems coming from other areas beyond combinatorics, such as algebraic geometry. This
has been done extensively by Meer [287, 288] , Bürgisser and Cucker [83, 84, 85] , Bürgisser,
Cucker and Lotz [89] , Scheiblechner [354, 355] (cf. [356] ), Bürgisser and Scheiblechner [95,
96] , Basu and Zell [42, 43] , and Basu [24] .
There are many counting problems in semialgebraic geometry (among them, (Ж) and
(З) ), but we will focus on (Д) and (Б) . The first one counts only components (i.e. “intrinsic
topological holes” of dimension 0 ), while the latter counts the “intrinsic topological holes”

10 Josué Tonelli-Cueto 0 §2
of all dimensions of a semialgebraic set. For more counting problems in real and complex
algebraic geometry, see [83, 84, 89] .
(Ж) : Size of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output #W ( p , Φ) ∈ Î ∪ { ∞ }
Complexity parameters s , q , D , n
Desir ed run-time s q n +1 D n · polylog ( n )
Known run-time s q n +1 D O ( n 2 ) [30, 31] [ (Д) + dimension]
(З) : Euler -Poincaré characteristic of a semialgebraic set.
Input p ∈ P d [ q ] , Boolean formula Φ over p of size ≤ s
Output Euler-Poincaré characteristic of W ( p , Φ) :
χ ( W ( p , Φ)) = ∑ i ( − 1) i β i ( W ( p , Φ))
Complexity parameters s , q , D , n
Desir ed run-time s q n +1 ( n D ) O ( n )
Known run-time s ( q n D ) O ( n ) [22, 186] 6
From the classical point of view (see [12, 308] for basic references), in which we restrict
the input of the problem to only integer polynomials, (Д) is FPSPACE -complete (with respect
to Cook-reductions 7 ) [322, 323, 103] and (Б) FPSPACE -hard [84] , this remains true if we
restrict to complex projective varieties [354] . However, this does not relate these problems
to the usual counting problems beyond the well-known inclusion # P ⊆ FPSPACE , and so
this only tells us that counting is harder in semialgebraic geometry than in combinatorics.
From the real point of view, as defined in [69, 68] , there is a real analogue of # P , # P Ò ,
which was introduced in [287, 288] and studied extensively in [84] . However, with this real
analogue, we can only show that (Б) is FP # P Ò -hard (due to the proof of [84 ; Theorem 7.1 ] ).
In [42, 43] , an alternative real analogoue of # P , # P †
Ò , was given. The main difference between
# P Ò and # P †
Ò is, roughly speaking, that the first one counts using the set-theoretic cardinal
and the second one using the sequence of Betti numbers. The most interesting part of this
real alternative analogue to # P is that it gives the following real version of Toda’s theorem
(cf. [389] ):
PH c
Ò ⊆ P # P †
Ò
Ò
where PH c
Ò is the compact version of PH Ò , meaning that we restrict the domain of each
block of quantified variables and of the block of free variables to the corresponding sphere.
To illustrate the above result in a weaker, but more accessible way, we consider, for
l > 0 , the decision problem GDP c
l whose instances are of the form
Q 1 x 1 ∈ Ó n 1 − 1 , . . . , Q l x l ∈ Ó n l − 1 , ( x 1 , . . . , x l ) ∈ W ( p , Φ)
6 The result in [186] is needed to extend the algorithm in [22] from closed to arbitrary semialgebraic sets.
However, it should be pointed that historically the substitution was done with the construction in [185] with the
proof of homotopy invariance given by [33, 35] . Nevertheless, the construction in [186] is the most general,
elegant and efficient one.
7 By a Cook-reduction from P to ˜
P, we will mean that there is a polynomial-time algorithm solving P with oracle
calls to ˜
P. When the restriction of polynomial-time is dropped, we will just say Turing-reduction .

0 §2 Condition and Homology in Semialgebraic Geometry 11
where n = ∑ l
i =1 n i , Q i ∈ { [ , \ } , p ∈ P d [ q ] and Φ a lax formula over p , which is a Boolean
formula without negations whose atoms are of the form ( f i = 0) , ( f i ≥ 0) and ( f i ≤ 0) .
This problem is the compact version of deciding quantified semialgebraic formulas with l
alternations, GDP l , where we quantify over Ò n i instead of Ó n i − 1 and we don’t put restrictions
on Φ . In [42, 43] , they showed that for fixed l > 0 , GDP c
l can be Cook-reduced (in the BSS
model of computation) to (Б) . In other words, (Б) is strong enough to decide (in an algebraic
model of computation) a compact version of the first-order theory of the reals. We note that
the opposite reduction is not possible (even if we just ask for a Turing-reduction), because
neither (Б) nor (Д) can be expressed in the first-order theory of the reals [48, 22] .
Putting together what we have said above, the problem (Б) is a hard problem to which
many hard problems in both the classical and real setting can be reduced. This problem is
intimately related to the real complexity class # P †
Ò , and a positive solution to it would bring
an inclusion into FEXPTIME Ò of many known problems and complexity classes in the world
of real complexity.
0 §2 -4 Historical 8
With the invention of Cartesian coordinates, geometric objects became formulas and
formulas geometric objects. This event, which marked the beginning of algebraic geometry,
allowed an “easy” algebraic understanding of many of the geometric objects of the past, such
as conics. However, with this new understanding, hordes of new “monstrous” algebraic-
geometric objects invaded the Greek classical world. Where once the harmony of Plato’s
shapes ruled, algebraic varieties created chaos with all their possible (real!) shapes. 9
This chaos took place even in the world of those algebraic varieties that we can draw
and see. At the beginning of the 20th century, there was a very modest success in classifying
the zoo of shapes of real algebraic curves and surfaces. The biggest successes were the un-
derstanding of curves of degree five (and partially those of degree six) by Harnack [215] and
Hilbert [222] and of surfaces of degree three by Schläfi [358] , Zeuthen [423] and Klein [259] .
In this context, in 1901, Hilbert formulated his famous 23 problems [224, 223] , and asked, in
the first half of the sixteenth one, how real algebraic smooth curves in Ð 2 and real algebraic
smooth surfaces in Ð 3 can look like (up to isotopy). The problem put special emphasis on
curves of degree six and surfaces of degree four.
Despite the early work in the problem by Ragsdale [320] and Rohn [337] , the main
collective effort went into other Hilbert’s problems. The solution to the seventeenth problem
led to the abstract theory of real fields by Artin and Schreier [13] . However, the main focus
went on the real version of Hilbert’s tenth problem, which is just nothing more than (Г) ,
motivated by the threat of undecidability created by the works of Gödel [192] , Church [127,
126] and Turing [394] . Fortunately, Tarsky [386] , 10 and later also Seidenberg [362] , showed
8 A warning is in order here. The history here will be told from a subjective perspective, starting in the 20th
century with Hilbert’s problems (specially the first half of Hilbert’s sixteenth problem [224, 223] ) and ignoring
the development of non-real algebraic geometry. The arrow of time in this history will be the understanding of
shapes (and topological invariants) of real algebraic and semialgebraic sets. For alternative narratives, see [157,
349, 32, 408] .
9 From time to time, the author will take these poetic licenses in this historical narrative to keep it entertaining.
10 Although published in 1951, Tarski’s work goes back to the 30s. The delay was due to the war.

12 Josué Tonelli-Cueto 0 §2
that the first-order theory of the reals was decidable. Unfortunately, the Second World War
brought the progress in the problem to a halt.
The Soviets 11 against Hilbert’ s sixteenth problem
With the war over, the only mathematical community that took the challenge of the first
half of Hilbert’s sixteenth problem was the Soviet one. And the narrative here has to be
complemented by the account of the Soviet protagonists themselves in [206] , [148] and
[408] .
In the 50s, continuing with the work of Petrovskii before the war [311] , Petrovskii and
Oleı  nik developed bounds for the Euler characteristic of real smooth algebraic sets [312] and,
later, Oleı  nik extended these results to sums of Betti numbers [303, 304] . This brought, for
the first time, a restriction on the possible topologies of general real algebraic sets.
In the 60s, Gudkov, following a suggestion of Petrovskii, attacked Hilbert’s sixteenth
problem for curves of degree six. After more that ten years and a famous mistake, 12 Gudkov
completed the classification of real projective algebraic curves of degree six [208, 209, 210] .
Meanwhile in the West, Milnor [292] and Thom [387] rediscovered the results of Oleı  nik on
the sum of Betti numbers [303, 304] , although providing new proofs which applied also to
the singular case.
In the 70s, the major breakthrough in the dissipation of the chaos in real algebraic ge-
ometry occurred, motivated by Gudkov’s congruence hypothesis [209] . In 1971, Arnold [7]
gave a proof of a weaker version of Gudkov’s conjecture using techniques from complex al-
gebraic geometry, in the flavour of Thom’s proof of Oleı  nik’s bound [387] and Klein’s proof of
Harnack’s inequality [260] . In a sudden boiling of ideas, Rokhlin developed this relation be-
tween real and complex algebraic geometry enormously. One year after Arnold, he gave the
first proof of Gudkov’s congruence hypothesis [341] ; 13 just four months afterwards, he gave
another simpler proof of a generalization of Gudkiv’s congruence hypothesis [340, 342] ; 14
and he completed this by exploring even further the relation between the topology of the real
part, its complexification and the relative position [343, 344, 345] .
With this explosion of ideas, the 70s and 80s were very successful for the Soviet school
of mathematics. The congruences and inequalities were generalized further by Fiedler, Gud-
kov, Kharlamov, Krakhnov, Nikulin and many others [179, 211, 247, 248, 249, 250, 299] .
These works on restrictions culminated with full classifications by Kharlamov, Nikulin and
Viro for new degrees at the end of the 70s and the beginning of the 80s. In the zoo of
curves, rigid 15 classifications were produced by Kharlamov for curves of degree five [253]
and by Nikulin for curves of degree six [298] ; and the isotopic classification for curves of
11 The term Soviet, instead of Russian, is necessary as not all Soviet mathematicians are Russian. For example,
Olga A. Oleı  nik was from Ukraine and Vladimir A. Rokhlin from Azerbaijan.
12 This mistake refers to the fact that the original classification of Gudkov in 1954 did not contain the curve
of type 1(5) ⨿ 5 , (one oval with five ovals inside and five outside). The later correction of this mistake was
surprising, because this type of curve was believed not to exist by Hilbert [224] . The discovery of this mistake
and its correction by Gudkov was possible thanks to Morozov [318] .
13 However, this proof had a mistake that it took eight years to be discovered and corrected by Marin [284] .
14 Funnily, of congruences mod 16 for Hilbert’s 16th problem
15 Rigid isotopy as opposed to topological isotopies, require that the isotopy can be carried out by deforming
the coefficient of the defining polynomials and not just the zero set.

0 §2 Condition and Homology in Semialgebraic Geometry 13
degree seven was obtained by Viro [411, 406, 405] . Further, the isotopic classification of
curves of degree eight was almost 16 completed by Viro [411, 406] , Shustin [371, 372, 373]
and Polotovskii [317] . In the zoo of surfaces, Viro gave constructions of M-surfaces in every
degree [401] . The partial results on (smooth) surfaces of degree four by Utkin [396, 395,
397, 207] were completed. The isotopic classification of surfaces of degree four was com-
pleted by both Kharlamov [246, 251, 252] and Nikulin [298] and the rigid classification by
Kharlamov [254] .
The emer gence of computational semialgebraic geometry
While the Soviet mathematical school was climbing the Everest that the first half of
Hilbert’s sixteenth problem still is, the Western schools of mathematics started to leave the
more pure approaches to semialgebraic geometry, as exemplified by [70] , to turn their atten-
tion into the existence of efficient algorithms in semialgebraic geometry. This was motivated
by the realization that, although in principle every problem expressible in the first-order the-
ory of the reals was solvable by [386, 362] , the algorithms by Tarski and Seidenberg would
probably solve the problem only after the universe was over, even for small size problems.
At the end of the 70s, these efforts condensed in the so-called Cylindrical Algebraic
Decomposition (CAD) developed independently by Collins [128] and Wüthrich [419] which
gave a complexity of O ( q D ) 2 O ( n ) to the decision of the first-order theory of the reals. There
were some hopes at the time that the lower bounds of Fischer and Rabin [181] could be
improved to show the optimality of CAD. 17 Around ten years later, Arnon, Collins and Mc-
Callum [9, 8] added some improvements to CAD; Ben-Or, Kozen and Reif [46] showed that
the computation could be performed in exponential space; and the lower bounds obtained
by Weispfenning [417] and Davenport and Heintz [145] made all the preceding work appear
optimal.
However, the Soviet school of mathematics still had one more surprise in store. On the
same year that the lower bounds by Weispfenning [417] and Davenport and Heintz [145]
appeared, Grigoriev and Vorobjov [203] (cf. [414, 413] ) and Grigoriev [202] (cf. [200] ) de-
veloped the critical points method , building on previous work in the same decade by Chis-
tov [125] , Chistov and Grigoriev [121, 122, 123, 205, 124] and Grigoriev [198, 199] on the
first-order theory of the complex numbers. In contrast to CAD, the run-time of the critical
points method is O ( q D ) n l where l is the number of quantifier alternations in the first-order
formula. This parameter was very present in the examples of [417, 145] .
In the coming decades, both CAD and the critical points method were successively
improved. CAD was improved by Hong [225, 226] , Collins and Hong [129] and many oth-
ers [108] . At the beginning of the 21st century, Brown [78] improved CAD for the plane, and
new examples were obtained by Brown and Davenport [79] which showed the importance
of the order chosen in CAD. The critical points method was improved by Canny [103, 105] ,
by Heintz, Roy and Solernó [219] , by Renegar [326, 327, 328] (cf. [325] ), where n l is sub-
stituted by ∏ i ( n i + 1) ; and finally by Basu, Pollack and Roy [26, 28] , where q and d are
separated and the exponent of q is given exactly without Landau notation. However, despite
16 Only the existence of six isotopy types remain to be resolved.
17 “The result of Fischer and Rabin suggests that a bound of this form is likely the best achievable for any
deterministic method”. [128]

14 Josué Tonelli-Cueto 0 §2
these incredible achievements, the main focus of computational real algebraic and semial-
gebraic geometry started to shift to the computation of topological invariants at the end of
the 80s.
“T uring” goes topological
At the end of the 80s, it was clear that the classical classification project of Hilbert was
difficult. After more than three decades of work, this project was completed for the cases
originally considered by Hilbert. However, the progress in terms of the degree was small: the
classification was only done for curves of degree at most seven and for surfaces of degree at
most four. It is not clear if this was the motivation, but around this time, after the successes
of CAD and the critical points method, a substantial amount of algorithms addressing the
computation of topological invariants emerged, especially concerning connected compo-
nents (Д) and the isotopy types of curves. 18
On (Д) , the progress started soon after the development of the critical points method.
The problem received an impetus from the applications thanks to the work of Canny [101]
showing the relation of the problem to robot motion planning. Soon after this, Canny [103,
102] developed the notion of a roadmap of a semialgebraic set that will play a fundamental
role. At the beginning of the 90s, a cluster of results showed that (Д) could be solved in
singly exponential time. Initially, Canny [107, 104] and Heintz, Krick, Roy and Solernó [218] 19
showed that, among other geometric-topological problems, deciding if two points belonged
to the same connected component could be done in singly exponential time. Then, almost
at the same time, Grigoriev and Vorobjov [412, 204] , Canny, Grigoriev and Vorobjov [106]
and Heintz, Roy and Solernó [218, 220] (see also [201] ) showed that (Д) could also be done
in singly exponential time. By the end of the 90s and beginning of the 2000s, the complexity
was improved to the more explicit O ( q D ) O ( n 2 ) by Basu, Roy and Pollack [27, 29, 30] . The
last significant progress in this problem was by Safey el Din and Schost [352, 353] and
Basu, Roy, Safey El Din and Schost [40] and Basu and Roy [39] at the beginning of the
2010s. They showed that for algebraic sets, the exponent O ( n 2 ) can be substituted by a
quasilinear factor in n , n · polylog ( n ) . As of today, extending this complexity bound to general
semialgebraic sets is seen as the biggest open problem in computational semialgebraic
geometry concerning (Д) .
On the isotopy type of real curves, there were some algorithms by Polotovkii [316] at
the end of the 70s and Gianni and Traverso [188] at the beginning of the 80s. However,
the first algorithm for smooth curves with a complexity estimate was given by Arnon and
McCallum [10, 11] relying on their previous work with Collins on CAD [9, 8] . For any kind of
curves, it was the algorithm by Roy and Szpirglas [348, 350] . The coming two decades saw
an improvement race of the algorithms and their complexity estimates by long sequence
of works: Cucker, González-Vega and Rosello [138] , Feng [178] , González-Vega and El
Kahoui [195] , González-Vega and Necula [196] , Eigenwillig, Kerber and Worper [171] , Ker-
ber [243] , Diochnos, Emiris and Tsigaridas [158] , Cheng, Lazard,Peñaranda, Pouget, Rouil-
lier and Tsigaridas [112] , Kerber and Sagraloff [244] , Diatta, Roullier and Roy [154] , Mehl-
18 It should be clear that such algorithms contribute to the goal of solving the first half of Hilbert’s sixteenth
problem in general and to the computational understanding of the topology of real algebraic varieties.
19 Unfortunately, the pelotita and the bolón didn’t get chosen as standard terminology in [34] .

0 §2 Condition and Homology in Semialgebraic Geometry 15
horn, Sagraloff and Wang [289, 290] , Kobel and Sagraloff [261] , and Diatta, Diatta, Rouillier,
Roy and Sagraloff [152] . All this sequence improved the complexity roughly from O ( d 23 ) to
O ( d 6 ) .
(Б) as the next step in this history
As we have seen above, with the coming of the new century, there were substantial im-
provements in algorithms solving (Д) and determining the isotopy type of curves. Regarding
Hilbert’s sixteenth problem, after the Soviet solution of the cases mentioned explicitly in the
problem, the progress has been more modest. However, the chaos of shapes of semialge-
braic geometry is far from becoming an understandable cosmos.
From the geometric perspective, the classification of curves of degree 8 is still open,
despite recent work by Chevallier [120] and Orevkov [306] ; new congruences, by Mikhalkin
[291] and Viro and Orevkov [291, 409] ; new assymptotics, by Orevkov and Kharlamov [307] ;
new examples, by Itenberg and Viro [411, 229, 230, 232] and Brugalle [80] . And, in the case
of surfaces, the situation is more dramatic, the Betti numbers are not even completely under-
stood: it is still not known whether there is a surface of degree five with 24 or 25 connected
components. The best example is until now by Bihan [55] and Orevkov [305] with 23 con-
nected components, which improved on the one before by Kharlamov and Itenberg with 22
components. Further, new limits on the existing construction techniques by Renaudineau
and Shaw [324] suggest that new ideas are needed.
From the computational perspective, the situation is more hopeful. On the one hand,
there were developments on both the algorithms for isotopy of curves, which we have already
discussed, and also some generalizations to surfaces and curves in 3-dimensional space of
these algorithms. On the other hand, the exploration on how to compute new Betti numbers
started.
At the beginning of the 2000s, algorithms were developed for the computation of topo-
logical invariants and piecewise linear approximations 20 of surfaces in space. The first al-
gorithms were developed by Fortuna, Gianni, Parenti and Traverso [184] , Fortuna, Gianni
and Luminati [182] , Cheng, Gao and Li [113] , Fortuna, Gianni, Luminati and Parenti [183] ,
and, at the end of the 2000s, the first one with a complexity analysis by Alberti, Mourrain
and Técourt [3] , which was based on previous work by Mourrain and Técourt [293] . The
situation for algorithms computing piecewise linear approximation of curves is similar. 21
In parallel to these developments, one should not ignore the developments coming
from other methods in computational geometry (cf. [71] ). At the beginning of the 90s, Sny-
der [376, 377] made substantial work in the isotopic piecewise-linear approximation of
curves. In the 2000s, the problem for curves and surfaces was dealt by Boissonnat, Cohen-
Steiner and Vegter [72, 73] , Plantinga and Vegter [315] , Stander and Hart [382] , Boissonat
and Oudot [74] , and Cheng, Dey, Ramos and Ray [115] . Although the focus of many of
these approaches was more on the correctness and applicability to general functions, not
only polynomials, many of these methods were fundamental in motivating developments in
20 This was necessary, because it is not clear, like it happens in the case of curves, which combinatorial
structure captures the topology of the istopy type of a surface. A torus can be knotted with itself.
21 See the works by Alcázar and Sendra, Gatellier, Labrouzy, Mourrain and Técourt [187] , El Kahoui [173] ,
Diatta, Mourrain and Ruatta [155] , and Cheng, Jin and Lazar [114] .

16 Josué Tonelli-Cueto 0 §3
computational semialgebraic geometry.
At the end of the 90s, Basu [21, 22] developed the first algorithm computing more
topological information (precisely (З) ) than just the 0th Betti number in singly exponential
time. By the middle of the 2000s, this was later extended to the first two Betti numbers
by Basu, Pollack and Roy [33, 35] and then to the first ℓ Betti numbers by Basu [23] .
However, in the last work, the complexity is doubly exponential in ℓ . In this period, one
should definitely mention that the constructions by Gabrielov and Vorobjov [185, 186] were
essential for extending algorithms from closed semialgebraic sets to general semialgebraic
sets.
In the last years, Basu and Riener [36, 37, 38] have applied successfully many of the
symbolic techniques to the case of symmetric semialgebraic sets described by symmetric
polynomials.
At this moment of historical development, (Б) is not just important because it is the last
step in a sequence of improvements, but because, as of today, (Б) resists to all technical
improvements coming from CAD and the critical points method. A solution to this problem, in
singly exponential time, would require new ideas in computational semialgebraic geometry.
These new ideas will give surely a better understanding of the topology of semialgebraic sets
and so contribute to the goal of the first half of Hilbert’s sixteenth problem.
0 §3 State-of-the-art and contributions to the problem
We will recall the state-of-the-art concerning (Б) and (В) and then explain our contribu-
tions to the problems, which we present in this thesis.
0 §3 -1 State-of-the-art regarding the problem
In the state-of-the-art regarding (Б) and (В) , we should distinguish between the symbolic
and numerical approaches. We will recall the existing symbolic methods for approaching (Б)
and their limits, and then explain the existing numerical method, the grid method, the dif-
ferences that it has with respect symbolic methods and what has been the progress so far
regarding (Б) and (В) .
Symbolic appr oaches
On the symbolic side, the best existing algorithm for (Б) is the algorithm by Basu [23] . It
computes the first ℓ Betti numbers of a semialgebraic sets in ( q D ) n O ( ℓ ) -time. This algorithm
is a product of a long sequence of steps, which can be read in the historical motivation,
but whose immediate ancestors are the algorithms by Basu [21, 22] and Basu, Pollack and
Roy [33, 35] and the constructions by Gabrielov and Vorobjov [185, 186] . We explain the
ideas present in these works.
Remark 0 §3 1 . In the case of complex smooth varieties, Scheiblechner [357] showed that
one can solve (Б) in singly exponential time and parallel polynomial time. The reason that
the techniques of [357] don’t apply to the corresponding version of (В) is because they are
based on de Rham cohomology which misses the torsion coefficients. However, it would be
interesting to study whether the techniques in [357] could be generalized to the real setting,
as they differ from the usual ones in computational semialgebraic geometry. ¶

0 §3 Condition and Homology in Semialgebraic Geometry 17
The first paper [21, 22] , by Basu, dealt with bounding the homology groups of semi-
algebraic sets. This was done by a certain covering and a Mayer-Vietoris argument that
allowed to bound the Betti numbers of the union in terms of the Betti numbers of the in-
tersections. Additionally, the techniques developed for the bound are used to compute the
Euler-Poincaré characteristic of semialgebraic sets (З) in singly exponential time. The main
idea for this was to make use of the addititivity of the Euler-Yao characteristic and the fact
that the Euler-Poincaré characteristic agrees with the Euler-Yao characteristic for closed
subsets. In this way, one could compute the Euler-Poincaré characteristic of S = ∪ σ S σ ,
with S σ not necessarily closed, using (under the right hypotheses) the identity
χ ( S ) = ∑
σ
χ ∗ ( S σ )
where χ is the Euler-Poincaré characteristic and χ ∗ is the Euler-Yao characteristic. In or-
der to compute it for each S σ , ideas coming from the critical points method and auxiliary
constructions are used.
The second paper [33, 35] , by Basu, Pollack and Roy, applied a similar strategy, to that
of [21, 22] , to compute the 0th and 1st Betti numbers. The main result is the construction,
in singly exponential time, of a cover by contractible semialgebraic sets. Then, in a variation
of the Nerve’s theorem, they show that for a closed set S = ∪ α S α with the S α closed
and contractible, one can compute β 0 ( S ) by counting the connected components of the
pairwise intersections of the S α , and β 1 ( S ) by counting the connected components of the
pairwise and triplewise intersections of the S α . This is proven using the so-called Mayer-
Vietoris double complex and its associated spectral sequence. Again, the computation of the
connected components (Д) of the possible intersections is done using the existing algorithms
for this problem (such as [30] ).
Additionally, in this paper, they gave the first proof that a general semialgebraic set can
be substituted by a closed semialgebraic with the same topological invariants. They did this
showing that the construction of Gabrielov and Vorobjov [185] also preserved the homotopy
type. This idea is fundamental, as almost all of the techniques with the name “Mayer-Vietoris”
only work with closed subsets. Interestingly, the construction [185] was developed to bound
the sum of the Betti numbers of general semialgebraic sets, which was an open problem at
the time.
The third paper [23] , by Basu, takes the algebraic-topological techniques in [33, 35]
to the limit. The main idea is to consider recursively contractible covers of the intersections
of elements of the previous covers. Then, using a suitable truncation of the Mayer – Vietoris
double complex and its associated spectral sequences, one is able to recover the first few
Betti numbers of the desired set. The doubly exponential explosion in the number of com-
puted Betti numbers is just a consequence of how the inductive construction of contractible
covers works.
One can see that in these works the main constructions are algebraic topological. The
main part of these algebraic topological arguments deal with how to get the Betti numbers
out of the constructed contractible cover (in [35] ) when the pairwise intersections are not
contractible. With the exception of the contractible cover in [35] , the main part of the ap-
proaches try solving (Б) by reducing, through algebraic topological computations, the prob-

18 Josué Tonelli-Cueto 0 §3
lem to cases that one can deal with by using the usual algorithms in semialgebraic geometry,
which mainly are algorithm dealing with (Д) and (Е) .
Numerical appr oaches
At the end of the 90s, a new approach of computation was introduced in real algebraic
geometry: the numerical one, to which the methods of this thesis belong. We introduce
below briefly what make the numerical algorithms numerical instead of symbolic. We also
give a short historical development as this was not included in the historical motivation 22
and we finish discussing the ideas of the grid method, which is the form that until now the
numerical approach has taken within real algebraic and semialgebraic geometry.
Numerical algorithms, condition numbers and condition-based complexity In
contrast to the symbolic approaches, the numerical approaches deal with inputs which are
assumed to be inexact and with which the performed operations perform inexactly. This
makes numerical algorithms different from symbolic ones, since it is possible to have ill-posed
inputs for which the numerical algorithm cannot give a correct answer (since for these inputs
arbitrarily small perturbations dramatically change the answer to the problem). In addition
to this, their complexity appears to depend not only on the size of the input, but also on a
parameter called condition number which measures the sensitivity of the problem, not the
algorithm, to variations in the input.
The condition-based complexity is a form of parameterized complexity in which the
focus is to understand, in terms of the condition number of the data, the complexity of the
numerical algorithm: number of operations used, precision needed… In this way, one can
understand why the algorithm works fast on certain inputs and slow or not at all on other
inputs.
However, one drawback is that we cannot know how the complexity depends, in gen-
eral, on the size of the input, which is necessary to compare the algorithm with symbolic
algorithms, for which this is usually the case. To solve this issue, the usual approach, going
back to Goldstine and von Neumann [194] , Demmel [150] and Smale [374] , is to consider a
“reasonable” probability distribution on the input-space and to study the probability distribu-
tion of the condition number. This will give a probabilistic complexity analysis that either holds
on average (average and smoothed complexity) or with high probability (weak complexity)
that is easier to compare with the worst-case complexity estimates of symbolic algorithms,
as no condition number appears in the bounds.
We will come back to these ideas in more detail in chapters 1 and 4 .
Some historical r emarks about the condition-based complexity paradigm The
condition-based complexity paradigm is not new and goes back to the beginning of the the-
ory of computation itself. In the middle of the 20th century, condition numbers were intro-
duced for the resolution of linear systems, respectively, by Turing [393] 23 and von Neumann
22 We should note that the use of an approach to a problem does mean that such approach is part of the
historical motivation of the problem. In other words, the historical motivation to approach numerically (Д) and (Е)
is not numerical.
23 See the notes of the 1970’s Turing Lecture by Wilkinson [418] for a first-person account of this discovery.

0 §3 Condition and Homology in Semialgebraic Geometry 19
and Goldstine [297] .
In the 80s, Demmel [149, 150] developed the framework of conic condition numbers
setting one of the most general frameworks of the condition-based complexity paradigm.
However, it was not until the 90s when condition-based complexity reached its age of ma-
jority by leaving the realm of numerical linear algebra. On the one hand, Shub and Smale [366,
367, 368, 370, 369] introduced condition numbers into the study of homotopy continuation
methods for solving complex systems of polynomials leading to the formulation of the so-
called Smale’s 17th problem [375] . The problem was solved successfully in the next two
decades by Beltrán and Pardo [45] , Bürgisser and Cucker [86] and Lairez [272] . On the
other hand, Renegar [329, 330, 331] introduced condition numbers into linear program-
ming. 24 This work was later further expanded by Cheung and Cucker [116, 117, 118] .
Together with the grid method, which we present below, the condition-based complex-
ity paradigm has become pervasive. Much of the current knowledge was condensed in the
book [87] by Bürgisser and Cucker, where a complete exposition of the main ideas of the
field and further historical comments can be found.
The grid method 25 In the realm of real algebraic geometry, condition numbers en-
tered for the first time through the works of Cucker and Smale [143, 144] and Cucker [133] .
They considered the problem of feasibility of a set of real polynomial equations. The basic
idea of the algorithm was to iteratively refine a grid until either one could certify the existence
of a zero, using Smale’s α -theory, or that the polynomial was either positive or negative,
using Lipschitz properties of the polynomial. Because of this, the numerical approach in real
algebraic geometry is called grid method . 26
The introduction of the grid method was motivated by the observation that the then-
existing symbolic methods were not likely stable due to the use of large matrices that had
to be inverted. 27 The way in which the grid method avoids this is by substituting inverting
large matrices by inverting a lot of small matrices, one at each point of the grid. Also, another
advantage of the grid method is that it is parallelizable, as one can perform the operations
at each point in the grid independently.
Ten years after, the grid method was further developed by Cucker, Krick, Malajovich
and Wschebor [139, 140, 141] . They apply it to count solutions of zero-dimensional real
polynomial systems. This problem is a particular case of (Ж) in which one restricts inputs
to polynomial systems with the same number of equations and variables. The idea of the
algorithm is as in [143, 144, 133] , to iteratively refine a grid until certain condition holds.
However, the main advancement, specially in [140, 141] , was a geometric interpretation of
the condition number in this setting [140] and a derived probabilistic analysis [140, 141] .
24 One should mention however the previous work of Goffin [193] that introduced condition numbers in a
limited setting of linear programming.
25 For comments on the grid method by one of its main characters, Cucker, see his surveys [134] and [135] .
26 We observe that the expression ‘grid method’ has a wide meaning. In this thesis, it will just refer to any
method based on the introduction of grids on a certain space and operations at each one of its points. Although
it should be clear that, in this sense, the grid method was definitely not invented by Cucker and Smale, its
application to real algebraic and semialgebraic geometry definitely was.
27 The correctness of this intuition was later confirmed by the theoretical results of Noferini and Townsed [302] .

20 Josué Tonelli-Cueto 0 §3
The interaction of the grid method and the computation of topological invariants (of
projective real algebraic sets) started with Cucker, Krick and Shub [142] . In [142] , they
incorporate techniques from topological data analysis such as the reach and the Niyogi-
Smale-Weinberger theorem [300 ; Proposition 7.1 ] , relating them to the existing condition
number of a real algebraic variety. Among the most interesting results in [142] is the first
explicit lower bound on the reach in terms of a parameter depending on the description of
the variety: Smale’s γ -invariant.
The generalization (from algebraic sets to basic semialgebraic sets) and simplification
of the grid method applied to the computation of homology was done by Bürgisser, Cucker
and Lairez [88] . In this work, sharper bounds and easier proofs were introduced for many
of the results in [142] and the results of topological data analysis in [300] . However, the
main progress occurred due to mainly two technical improvements: a bound for the reach
of an intersection in terms of the reach of intersections of the boundaries, and a method
for choosing the approximating points in the grid based on an inclusion-criterion and not an
inclusion-exclusion scheme.
The main similarity between the above numerical algorithms with the symbolic ones
in [33, 35, 23] is that the homology is computed by computing a cover of the set. The main
difference lies in the fact that the covers in [33, 35, 23] are difficult to describe (each element
is a semialgebraic set with its own description) and they are not topologically nice (i.e., they
do not satisfy the Leray property), while the covers produced by the grid method [142, 88]
are easy to describe (they are just a union of balls) and are topologically nice (i.e. they satisfy
the Leray property). This property of the covers produced by the grid method is what allows
numerical algorithms to use easier algebraic topological tools to compute the homology and
so to be faster.
0 §3 -2 Contributions to the problem
Our contributions to the problem are, mainly, to produce both algebraic topological and
semialgebraic tools to compute the homology of semialgebraic sets using the grid method,
which allows to give the first algorithm that solves (В) (and so also (Б) ) in singly exponential
time with high probability. This was done in [91, 92] together with Bürgisser and Cucker. An-
other contribution, which belongs to the different problem of computation of piecewise linear
isotopic approximations, is to provide a complexity analysis of an algorithm, the Plantinga-
Vegter algorithm. We show that this algorithm works in average polynomial time. The impor-
tance of this result lies in the fact that it opens the door, for the first time ever, to numerical
algorithms based on the grid method to be shown to run in finite expected time. This was
done in [136] together with Cucker and Ergür.
(В) in weak singly exponential time
In [91] , the main addition to the grid method was the development of a method to
construct a simplicial complex with the same homology as a closed general semialgebraic
set. This was done by developing functorial methods in topological data analysis. With this
method, we can construct the simplicial complexes just by constructing it for the atoms and
then combining the “atomic simplicial complexes” in the same manner the formula of the
semialgebraic set combines the atoms. Another crucial step was a quantitative version of

0 §3 Condition and Homology in Semialgebraic Geometry 21
Durfee’s theorem [163] .
In [92] , the main contribution was to develop a quantitative version of the Gabrielov-
Vorobjov theorem [186] . In this quantitative version, the original inequalities ≪ , meaning
sufficiently small in a precise sense, were substituted by simple strict inequalities of the form <
and a global upper bound linear in the inverse of the condition number. Here, the application
of the Mather-Thom theory introduced in [91] is necessary. With this explicit version, one
can just apply the construction of Gabrielov and Vorobjov [186] to pass from the general
case to the closed case.
One can see a certain similarity with the symbolic history. In a way, the results in [91]
are analogous to those in [23] in the sense that both deal with how to get more topological
information out of covers. The reason why [91] leads to a better run-time with high proba-
bility is that the covers of the grid method are simpler and so they can be combined in an
easier way than those used before in [23] . Similarly, the core results of [92] , the quantitative
Gabrielov-Vorobjov theorem, are completely analogous to that used in [33, 35] to pass from
the general to the closed case. The main difficulty for the grid method is that we cannot
leave the realms of the real numbers, while in the symbolic methods the fact that the original
inequalities work only for sufficiently small numbers is not relevant algorithmically as one can
work with infinitesimals to go around this issue.
Despite all possible similarities, the underlying methods of the symbolic algorithms in [21,
33, 35, 23] are fundamentally different from those of the grid method. This is the main reason
behind the progress in [91, 92] that has brought (В) down to weak singly exponential time.
These ideas will be exposed in Chapters 1 , 2 , 3 and 4 .
Adaptive grid method
The motivation for the work in [136] was the observation that a condition-based com-
plexity analysis of the Plantinga-Vegter algorithm [315] , which computes an isotopic piece-
wise-linear approximation of implicit curves and surfaces, would be possible. The existing
complexity analysis by Burr, Gao and Tsigaridas [98, 99] gave only complexity bounds expo-
nential on the degree. The progress in [136] relied on a condition-based approach and the
continuous amortization technique developed by Burr, Krahmer and Yap [100] and Burr [97] .
Interestingly, subdivision-based methods such as the Plantinga-Vegter algorithms can
be interpreted as adaptive grid methods. The main difference with the usual grid method is
that the grid is not refined globally, but locally depending on whether at a given point we
need a finer grid. In this way, the main contribution in [136] was that, for the first time, it
showed that an algorithm in real numerical algebraic geometry could have finite expected
time. Moreover, the probabilistic estimates of [136] didn’t come from integral geometry, but
from geometric functional analysis like those or Ergür, Paouris and Rojas [175, 176] . This
allowed for a probabilistic bound using probability distributions more general than the normal
distribution.
However, the most important contributions in [136] are in the future possibilities that it
suggests. On the one hand, it shows how the condition-based complexity can be applied
to analyze subdivision-based methods. On the other hand, it has opened a roadmap to the
development of a numerical algorithm that not only solves (В) in weak singly exponential
time, but in average singly exponential time. We explore this possibility in Chapter 5 .

22 Josué Tonelli-Cueto AI
AI Analytical index
The following analytical index briefly presents the content of each chapter. It is intended
to give an overview of the topics and structure of the thesis.
Chapter 1
The condition number for problems involving semialgebraic sets measures how transversely
the zero sets of the polynomials defining it intersect. In the special case of spherical alge-
braic sets, this condition number is well-behaved and has good properties, both geometric
and probabilistic. In the general case, this properties are transmitted in the homogeneous
case, almost immediately, and in the affine case, after some effort. A probabilistic analysis
of each of these notions of condition numbers is performed for normally distributed random
polynomials.
Chapter 2
When the condition number is finite, certain deformations of the semialgebraic set can be
done with explicit constants and inequalities, depending linearly on the inverse of the con-
dition number itself. The main tools are differential topology and Mather-Thom theory. For
well-posed cases, quantitative versions are given for Durfee’s theorem and the Gabrielov-
Vorobjov construction.
Chapter 3
The homology of a closed set can be computed using clouds of points (i.e., unions of balls).
A measure for the quality of the approximation is the Hausdorff distance. There is a geomet-
ric property of the set, the reach or local feature size, which controls the size of correctly-
approximating clouds of points (Niyogi-Smale-Weinberger theorem). The reach behaves well
with respect to intersections, analytic and basic semialgebraic subsets. The homology of
a cloud of points can be computed by considering only the intersection relations of the
cover (Nerve theorem). It is enough to consider the pairwise intersections of the cover (Attali-
Lieutier-Salinas theorem).
Chapter 4
Numerical algorithms are a valuable tool for solving problems. There is an algorithm for com-
puting the homology of semialgebraic sets which is singly exponential with high probability.
We do a condition-based and a probabilistic complexity analysis of this algorithm. The nu-
merical algorithm have parts that are highly parallelizable and it is numerically stable.
Chapter 5
One can perform the probabilistic analysis for the condition number of random polynomi-
als that are not normally distributed. Adaptive methods can provide faster numerical algo-
rithms: this is shown with the Plantinga-Vegter algorithm and the Han covering algorithm.
We propose a пятилетка program with questions and problems regarding the computation
of topological invariants and the first half of Hilbert sixteenth problem.

AI Condition and Homology in Semialgebraic Geometry 23
Appendix F
The properties of the real zero set of a real system of polynomials is governed by the number
of non-zero terms in the system and not necessarily the degree of the terms. The classical re-
sults on fewnomial are still far from the resolution of Kushnirenko’s hypothesis. Kushnirenko’s
hypothesis is true on average. Fewnomial systems with very few terms have with very high
probability no real zeros. Problems related to a possible probabilistic theory of fewnomials
are stated.
Apéndice M
El tema principal de esta tesis es el cálculo numérico de grupos de homología de conjuntos
semialgebraicos. Yendo término por término, tratamos de dar una imagen global de cuál es
el tema a una persona que no esté familiarizada con las matemáticas.

25
There was nothing there now except a single Commandment. It ran:
ALL ANIMALS ARE EQUAL
BUT SOME ANIMALS ARE MORE EQUAL
THAN OTHERS.
George Orwell, Animal Farm: A Fairy Story
1
Condition numbers
for the homology of semialgebraic sets
Like the animals in George Orwell’s Animal Farm [Q11] , all inputs of a given size are
equal, but some inputs are more equal than others for a numerical algorithm. The condition
number, which is a measure of numerical sensitivity with respect to the problem, lies at the
hearth of this difference.
In general, a condition number of an input with respect to a problem measures how
much the answer to the problem changes depending on how much the input changes.
However, we are dealing with problems such as (Б) and (В) where the output is discrete.
Because of this, in these problems, the condition number should bound the inverse of max-
imum possible variation of the input such that the output does not change.
In this chapter, we will introduce the condition number κ aff that will be the basis of our
condition-based complexity analyses. To do this, we deal first with the homogeneous and
spherical case where the usual condition-based framework for real algebraic geometry has
been developed.
First, we introduce the Weyl norm, which will be the “ruler” we will use to measure
the variations in the space of polynomials, and the class of KSS random polynomial tuples;
second, we introduce the condition number of an homogeneous polynomial tuple; third,
we introduce the intersection condition number of an homogeneous polynomial tuple; and
fourth and last, we introduce the intersection condition number of a polynomial tuple in the
non-homogenous case. In all the cases, we will discuss the properties and deduce the cor-
responding bounds, deterministic and probabilistic, of all the introduced condition numbers.

26 Josué Tonelli-Cueto 1 §1
1 §1 Homogeneous polynomials and Weyl norm
Fix q , n ∈ Î and d = ( d 1 , . . . , d q ) ∈ Î q . Consider the space of d -homogeneous
polynomial q -tuples
H d [ q ] := { f ∈ Ò [ X 0 , X 1 , . . . , X n ] q | f i homogeneous of degree d i } ( 1.1 )
and let D := max i ∈ [ q ] d i . Let X α := X α 0
0 · · · X α n
n and write every f i as f i = ∑ | α | = d i f i , α X α , so
that f i , α denotes the α -coefficient of f i . The Weyl norm is the norm given by
∥ f ∥ W := v
u
t q
∑
i =1 ∥ f i ∥ 2
W and ∥ f i ∥ W := v
u
t ∑
| α | = d i ( d i
α ) − 1
f i , α ( 1.2 )
where ( d i
α ) = d i !
α 0 ! α 1 ! ··· α n ! is the multinomial coefficient. Note that the Weyl norm comes from
an inner product, which we will write as
⟨ f , g ⟩ W =
q
∑
i =1 ∑
| α | = d i ( d i
α ) − 1
f i , α g i , α ( 1.3 )
for f , g ∈ H d [ q ] .
Remark 1 §1 1 . Although the definition above and the results below are stated over the real
numbers, the analogous results hold over the complex numbers, when we substitute the the
Weyl norm by its complex version. This will only be important when we prove Lemma 1 §2 8 ,
whose easiest proof is by passing through the complex version of the results here. ¶
1 §1 -1 The three main properties of the Weyl norm
There are three reasons why the Weyl norm is used. First, it allows one to write nice
formulas for the point-wise evaluation and derivation of polynomials. Second, it has an or-
thogonal invariance, which means it does not favor any direction in the space where we
evaluate homogenous polynomials. Third, it controls the norm of the evaluations of the poly-
nomials and their derivatives.
Evaluation and derivation as polynomials
Let x , v ∈ Ó n be such that v ∈ T x Ó n and i ∈ [ q ] , then define the polynomial tuples
ev i
x := ⟨ x , X ⟩ d i e i ∈ H d [ q ] and dev i
x , v := √ d i ⟨ x , X ⟩ d i − 1 ⟨ v , X ⟩ e i ∈ H d [ q ] ( 1.4 )
where e i ∈ Ò q is the vector with one in the i th component and zeros in the rest. We view X
as the vector ( X 0 , . . . , X n ) ∗ and we write ⟨ · , · ⟩ for the standard inner product of Ò n .
Pr oposition 1 §1 1 . Let f ∈ H d [ q ] and x , v ∈ Ó n such that v ∈ T x Ó n . Then
⟨ f , ev i
x ⟩ W = f i ( x ) and ⟨ f , dev i
x , v ⟩ W = 1
√ d i
D x f i v .
In particular, ∥ e v i
x ∥ W = 1 , ∥ dev i
x , v ∥ W = 1 and ⟨ ev i
x , dev i
x , v ⟩ W = 0 .

1 §1 Condition and Homology in Semialgebraic Geometry 27
Proof. By the multinomial formula,
ev i
x = ( n
∑
k =0
x i X i ) d i
e i = ©  « ∑
| α | = d i ( d i
α ) x α X α ª ® ¬ e i
and so, by ( 1.3 ),
⟨ f , ev i
x ⟩ W = ∑
| α | = d i
f i , α x α = f i ( x ) .
Similarly, one can see that
dev i
x , v = 1
√ d i ©  « ∑
| α | = d i ( d i
α ) ( n
∑
k =0
α k x α − e k v k ) X α ª ® ¬ e i ,
and so, by ( 1.3 ),
⟨ f , dev i
x , v ⟩ W
= 1
√ d i ∑
| α | = d i
f i , α ( n
∑
k =0
α k x α − e k v k ) = 1
√ d i
n
∑
k =0 ©  « ∑
| α | = d i
α k f i , α x α − e k ª ® ¬ v k = 1
√ d i
D x f i v ,
where the last equality holds because v is orthogonal to x . The last claim follows easily from
computing the evaluation and the derivative. □
Cor ollary 1 §1 2 . The set { ev i
x | x ∈ Ó n , i ∈ [ q ] } linearly spans H d [ q ] .
Proof. By Proposition 1 §1 1 , the orthogonal complement of this set is
{ f ∈ H d [ q ] | for all x ∈ Ó n , f ( x ) = 0 } .
So we only have to show that this is the zero subspace. However, the only homogenous
polynomial which vanishes in all points of the sphere is the zero polynomial. Therefore the
claim follows. □
Cor ollary 1 §1 3 . [87 ; §16.3 ] . For every x ∈ Ó n , there is an orthogonal decomposition
H d [ q ] = C x ( H d [ q ]) ⊕ L x ( H d [ q ]) ⊕ R x ( H d [ q ])
where
(R) R x ( H d [ q ]) := { g ∈ H d [ q ] | g ( x ) = 0 , D x g = 0 } ,
(L) L x ( H d [ q ]) := { g ∈ R x ( H d [ q ]) ⊥ | g ( x ) = 0 } = span ( dev i
x , v | i ∈ [ q ] , v ∈ Ó n ∩ x ⊥ ) ,
and
(C) C x ( H d [ q ]) := { g ∈ R x ( H d [ q ]) ⊥ | D x g = 0 } = span ( ev i
x | i ∈ [ q ]) .
Further, { ev i
x } i ∈ [ q ] is an orthonormal basis of C x ( H d [ q ]) and, for { v j } j ∈ [ n ] an orthonormal
basis of T x Ó n = x ⊥ , { ev i
x , v j } i ∈ [ q ] , j ∈ [ n ] is an orthonormal basis of L x ( H d [ q ]) .

28 Josué Tonelli-Cueto 1 §1
Proof. Fix an orthonormal basis { v j } j ∈ [ n ] of x ⊥ . Then, by Proposition 1 §1 1 , we have that
{ ev i
x , dev i
x , v 1 , . . . , dev i
x , v x } i ∈ [ q ] is an orthonormal system. Now, also by Proposition 1 §1 1 ,
we have that R x ( H d [ q ]) is its orthogonal complement, and so
R x ( H d [ q ]) ⊥ = span ( ev i
x , dev i
x , v 1 , . . . , dev i
x , v x | i ∈ [ q ]) .
Thus C x ( H d [ q ]) , L x ( H d [ q ]) ⊆ span ( ev i
x , dev i
x , v 1 , . . . , dev i
x , v x | i ∈ [ q ]) . Now, by Proposi-
tion 1 §1 1 , we have that
C x ( H d [ q ]) ⊥ span ( dev i
x , v j | i ∈ [ q ] , j ∈ [ n ]) and L x ( H d [ q ]) ⊥ span ( ev i
x | i ∈ [ q ])
and
C x ( H d [ q ]) ⊇ span ( ev i
x | i ∈ [ q ]) and L x ( H d [ q ]) ⊇ span ( dev i
x , v j | i ∈ [ q ] , j ∈ [ n ]) .
This concludes the proof. □
Remark 1 §1 2 . When H d [ q ] is clear from the context, we will just write C x , L x and R x instead
of C x ( H d [ q ]) , L x ( H d [ q ]) and R x ( H d [ q ]) . ¶
Orthogonal invariance
The natural action of O ( n + 1) on Ò n extends naturally to an action on H d [ q ] given by
pre-composition. Given f ∈ H d [ q ] and u ∈ O ( n + 1) , we define
f u := f ( u X ) ( 1.5 )
where u X is the multiplication of the vector X = ( X 0 , . . . , X n ) ∗ with u . Note that this means
that we are viewing the action of O ( n + 1) on H d [ q ] as a right action.
Pr oposition 1 §1 4 . [87 ; Theorem 16.3 ] . Let f , g ∈ H d [ q ] and u ∈ O ( n + 1) . Then
⟨ f u , g u ⟩ W = ⟨ f , g ⟩ W and ∥ p u ∥ W = ∥ p ∥ W .
Proof. It is enough to prove that for some generating subset S ⊆ H d [ q ] , the claim holds for
all f , g ∈ S. Let S be the generating set of Corollary 1 §1 2 . For all ev i
x ∈ S and u ∈ O ( n + 1) ,
( ev i
x ) u = ev i
u − 1 x . Therefore for all ev i
x , ev j
y ∈ S and u ∈ O ( n + 1) ,
⟨ ( ev i
x ) u , ( ev j
y ) u ⟩ W = ⟨ ev i
u − 1 x , ev j
u − 1 y ⟩ W .
If i , j , this equals zero. If i = j , then, by Proposition 1 §1 1 , it equals
⟨ u − 1 x , u − 1 y ⟩ d i = ⟨ x , y ⟩ d i
where the equality follows from the fact that u is orthogonal. Hence for every ev i
x , ev j
y ∈ S and
u ∈ O ( n + 1) , the value of ⟨ ( ev i
x ) u , ( ev j
y ) u ⟩ W is independent of u and so equals ⟨ ev i
x , ev j
y ⟩ W ,
as desired. □
Remark 1 §1 3 . We note that, in contrast with the unitary action on complex homogeneous
polynomials, H d [1] is not an irreducible O ( n + 1) -module. This means that the Weyl norm
is not the unique, up to scalar multiplication, orthogonally invariant norm on H d [1] . More-
over, one can check that the Weyl norm on H d [1] is not equal to the L 2 -norm, ∥ p ∥ 2 :=
√ Å x ∈ Ó n ∥ p ( x ) ∥ 2 , up to any scalar, because while any two monomials are orthogonal with
respect to the Weyl inner product, this is not true with respect to the L 2 inner product with
the exception of the linear case. ¶

1 §1 Condition and Homology in Semialgebraic Geometry 29
Cor ollary 1 §1 5 . [87 ; §16.3 ] . Let x ∈ Ó n and consider the orthogonal decomposition of
H d [ q ] of Corollary 1 §1 3 . Then for all u ∈ O ( n + 1) ,
C u
x = C u − 1 x , L u
x = L u − 1 x and R u
x = R u − 1 x .
In particular, the orthogonal decomposition of H d [ q ] of Corollary 1 §1 3 remains invariant
under those orthogonal transformations that fix x .
Proof. This is immediate from the definitions of Corollary 1 §1 3 and the fact that the action
of O ( n + 1) on H d [ q ] respects the Weyl inner product, by Proposition 1 §1 4 . □
Evaluation and derivative bounds
Recall that any space of matrices Ò a × b has an inner product, called Frobenius inner
product , given by
⟨ M , ˜
M ⟩ F := tr ( M ˜
M ∗ ) ( 1.6 )
for M , ˜
M ∈ Ò a × b . Associated to this inner product, we have the Frobenius norm ∥ M ∥ F .
Another possible norm is the operator norm which is given by
∥ M ∥ = max
v ∈ Ó n ∥ M v ∥ ( 1.7 )
Recall that for all M ∈ Ò a × b , ∥ M ∥ ≤ ∥ M ∥ F .
Pr oposition 1 §1 6 . Let x ∈ Ó n . Consider the linear map
R x : H d [ q ] → Ò q × ( n +2)
f 7→ ( R 0
x ( f ) R 1
x ( f ) ) := ( f ( x ) ∆ − 1
d D x f ( É − x x ∗ ) )
where
∆ d = ©   «
√ d 1
. . . √ d q ª ® ® ¬
. ( 1.8 )
Then R x is an orthogonal projection whose image is given by
im R x = { ( z , M ) ∈ Ò q × Ò q × ( n +1) | M x = 0 } .
Proof. Consider an orthonormal basis { v l } l ∈ [ n ] of x ⊥ . From this basis, one can construct
{ e k , e k v ∗
1 , . . . , e k v ∗
n } k ∈ [ q ] which is an orthonormal basis of { ( z , M ) ∈ Ò q × Ò q × ( n +1) |
M x = 0 } . In the coordinates of this latter basis, by Proposition 1 §1 1 , we have that R x ( f ) is
written as
( ⟨ ev 1
x , f ⟩ ⟨ dev 1
x , v 1 , f ⟩ · · · ⟨ dev 1
x , v n , f ⟩ · · · ⟨ ev q
x , f ⟩ ⟨ dev q
x , v 1 , f ⟩ · · · ⟨ dev q
x , v n , f ⟩ ) ∗ .
Since { ev k
x , dev k
x , v 1 , . . . , dev k
x , v n } k ∈ [ q ] is an orthonormal system by Proposition 1 §1 1 , the
claim follows. □
Cor ollary 1 §1 7 . Let f ∈ H d [ q ] and x ∈ Ó n . Then
∥ f ( x ) ∥ ≤ ∥ f ∥ W and ∥ ∆ − 1
d D x f ∥ ≤ ∥ f ∥ W
where ∥ · ∥ is the operator norm and ∆ d as in ( 1.8 ) .

30 Josué Tonelli-Cueto 1 §1
Proof. R 0
x : f 7→ f ( x ) is an orthogonal projection, by Proposition 1 §1 6 . Therefore the
first inequality holds. For all v ∈ T x Ó n ∩ Ó n , f 7→ ∆ − 1
d D x f v is an orthogonal projection,
because, by Proposition 1 §1 6 , it is a composition of orthogonal projections. Therefore the
second inequality follows. □
Recall that the geodesic distance on Ó n , dist Ó , is the distance given
dist Ó ( x , ˜
x ) := arccos ⟨ x , ˜
x ⟩ . ( 1.9 )
One can see that dist Ó ( x , ˜
x ) is the length of the shortest path inside Ó n joining x and ˜
x ,
which is why it is called “geodesic”.
Cor ollary 1 §1 8 (Exclusion lemma). Let f ∈ H d [ q ] and x , ˜
x ∈ Ó n . Then
∥ f ( x ) − f ( ˜
x ) ∥ ≤ √ D ∥ f ∥ W dist Ó ( x , ˜
x )
where dist Ó is the geodesic distance on Ó n .
Proof. Let γ : [0 , 1] → Ó n be a constant speed geodesic going from x to ˜
x . Then
∥ f ( x ) − f ( ˜
x ) ∥ =     ∫ 1
0
D γ ( t ) f γ ′ ( t ) d t     ≤ dist Ó ( x , ˜
x ) ∫ 1
0 ∥ D γ ( t ) f ∥ d t ,
where we used that ∥ γ ′ ( t ) ∥ = dist Ó ( x , ˜
x ) for a constant speed geodesic whose domain
is the interval [0 , 1] . We have ∥ D γ ( t ) f ∥ ≤ √ D ∥ ∆ − 1
d D γ ( t ) f ∥ ≤ √ D, due to the bound in
Corollary 1 §1 7 . This concludes the proof. □
However, we can prove an stronger version of the above corollary where we bound the
operator norm of all derivatives. For it, we need to sharpen the bound of the operator norm of
D x f in order to be able to apply an inductive argument. Recall that D x f denotes the tangent
map of f as a function on Ò n +1 , while D x f the tangent map as a function on Ó n .
Pr oposition 1 §1 9 . Let f ∈ H d [ q ] and v ∈ Ó n . Then D X f v ∈ H d − 1 [ q ] and
∥ D X f v ∥ W ≤ D ∥ f ∥ W ,
where ∆ d is as in ( 1.8 ) . Further, ∥ ∆ − 1
d D X f v ∥ W ≤ √ D ∥ f ∥ W .
Proof. By Proposition 1 §1 4 , we can assume without loss of generality that v = e 0 . Indeed,
let u ∈ O ( n + 1) be such that u e 0 = v , then
∥ ∆ − 1
d D X f v ∥ W = ∥ (∆ − 1
d D X f u e 0 ) u ∥ W = ∥ ∆ − 1
d D X f u e 0 ∥ W ≤ ∥ f u ∥ W = ∥ f ∥ W .
Let M i
d , α := ( d i
α ) 1
2 X α e i . Then { M i
d , α | i ∈ [ q ] , | α | = d i } is an orthonormal basis of
H d [ q ] . By direct computation,
⟨ M i
d − 1 , α , D X f e 0 ⟩ W = ⟨ d i M i
d − 1 , α X 0 , f ⟩ W .
Hence the linear map L : H [ q ] → H d − 1 [ q ] given by f 7→ D X f v can be written as
∑
i , α
M i
d − 1 , α ( d i M i
d − 1 , α X 0 ) ∗ .

1 §1 Condition and Homology in Semialgebraic Geometry 31
Now, { d i M i
d − 1 , α X 0 | i ∈ [ q ] , | α | = d i − 1 } is an orthogonal system, although not an orthonor-
mal one, such that for each i and α , ∥ d i M i
d − 1 , α X 0 ∥ ≤ d i ≤ D. Therefore ∥ L ∥ ≤ D and so the
main claim follows. For the last claim, note that ⟨ d i M i
d − 1 , α X 0 , ∆ − 1
d f ⟩ W = ⟨ √ d i M i
d − 1 , α X 0 , f ⟩ W
and proceed analogously using the latter expression. □
Recall that for a k -multilinear map L : Ò n 1 +1 × · · · × Ò n k +1 → Ò q , its operator norm
is given by
∥ L ∥ := max { ∥ L ( v 1 , . . . , v k ) ∥ | v 1 ∈ Ó n 1 , . . . , v k ∈ Ó n k } .
Cor ollary 1 §1 10 . Let f ∈ H d [ q ] and x ∈ Ó n . Then, for all k ≥ 1 ,
    1
k ! ∆ − 1
d D k
x f     ≤ 1
√ D ( D
k ) ∥ f ∥ W
where ∥ · ∥ is the operator norm for multilinear maps.
Proof. Fix v 1 , . . . , v k ∈ Ó n , then
    1
k ! ∆ − 1
d D k
X f ( v 1 , . . . , v k )     W
= 1
k !    ∆ − 1
d D k
X f ( v 1 , . . . , v k )    W
≤ D − k
k !    ∆ − 1
d D k − 1
X f ( v 1 , . . . , v k − 1 )    W ≤ · · · ≤ ( D − k ) · · · ( D − 2)
k !    ∆ − 1
d D X f ( v 1 )    W
≤ ( D − k ) · · · ( D − 2) √ D
k ! ∥ f ∥ W = 1
√ D ( D
k ) ∥ f ∥ W
by applying inductively the (first) inequality of Proposition 1 §1 9 in the second line and the last
inequality in the last line. Then Corollary 1 §1 7 and maximising over v 1 , . . . , v k finishes the
proof. □
1 §1 -2 Random polynomials in H d [ q ]
Among all distributions in Ò N , there is one that occupies a special place: the normal
distribution. Recall that the normal distribution centered at x ∈ Ò N and with standard devi-
ation σ > 0 , N ( x , σ ) , is the absolutely continuous probability distribution on Ò n with density
function
z 7→ 1
(2 π ) N /2 σ N e − ∥ z − x ∥ 2
2 σ 2 .
The standard normal distribution is the normal distribution centered at 0 and with typical
deviation 1 . A Gaussian random vector x ∈ Ò N is a random vector distributed according to
the standard normal distribution N (0 , 1) . To indicate that a random vector x ∈ Ò N has the
normal distribution N ( x , σ ) , we will simply write x ∼ N ( x , σ ) .
This distribution has many properties that make it special:
1. It is invariant under orthogonal transformations, and so it does not favor any direction
in space. I.e., if x is a random vector such that x ∼ N (0 , σ ) and u ∈ O ( N ) , then
u x ∼ N (0 , σ ) .

32 Josué Tonelli-Cueto 1 §1
2. If x ∈ Ò N and y ∈ Ò N ′ are independent random vectors such that x ∼ N ( x , σ ) and
y ∼ N ( y , σ ) , then ( x
y ) ∼ N ( ( x
y ) , σ ) .
3. If x , y ∈ Ò N are independent random vectors such that x ∼ N ( x , σ ) and y ∼ N ( y , ς )
and λ , µ ∈ Ò , then λ x + µ y ∼ N ( λ x + µ y , √ λ 2 σ 2 + µ 2 ς 2 ) . In particular, x + y ∼
N ( x + y , √ σ 2 + ς 2 ) .
4. If x ∈ Ò N is a random vector such that x ∼ N ( x , σ ) and P : Ò N → Ò N ′ is an orthogonal
projection, then P x ∼ N ( P x , σ ) .
5. If x ∈ Ò N is a Gaussian random vector, then x / ∥ x ∥ has the uniform distribution on the
unit sphere Ó N − 1 , U ( Ó N − 1 ) .
6. Among all probability distributions on Ò N with the same mean and covariance matrix 1 ,
the normal distribution is the one having maximum entropy [132 ; Theorem 9.6.5 ] . In
other words, it is the distribution to choose when only the mean and covariance matrix
of a distribution are available.
In a finite dimensional real vector space with an inner product, we can define analogously
the normal distribution N ( x , σ ) by considering the corresponding norm instead. This will allow
us to talk about Gaussian random matrices where the inner product is the Frobenius one and
about random polynomials, by considering the Weyl inner product of H d [ q ] . This motivates
the following definition.
Definition 1 §1 1 . A KSS random polynomial tuple f is a random polynomial tuple f ∈ H d [ q ]
with an absolutely continuous distribution whose density function is given by
δ f ( f ) := 1
(2 π ) N /2 e − ∥ f ∥ 2
W
2 .
Remark 1 §1 4 . The terms “KSS” is an acronym for Kostlan-Shub-Smale, in honor of the
creators of the distribution. See [265] , [166] and [367] . ¶
We finish with a proposition that will be useful later, since it gives the probability distri-
bution of the norm of a KSS random polynomial tuple. Recall that the χ 2 -distribution with m
degrees of freedom, χ 2
m , is the probability distribution of the square of the norm of a Gaussian
random vector x ∈ Ò m . One can easily show that its density function is given by
1
2 m /2 Γ( m /2) t m
2 − 1 e − t
2
for t > 0 , where Γ( s ) := ∫ ∞
0 x s − 1 e − x d x is Euler’s Γ function.
1 The covariance matrix of a random vector x ∈ Ò N is the matrix Å ( x − Å x ) ∗ ( x − Å x ) . If x ∼ N ( x , σ ) , then σ 2 É
is the covariance matrix of x .

1 §2 Condition and Homology in Semialgebraic Geometry 33
Pr oposition 1 §1 11 . Let x ∈ Ò m be a Gaussian random vector. Then the density function
of ∥ x ∥ is given by
δ ∥ x ∥ ( t ) = 1
2 m /2 − 1 Γ( m /2) t m − 1 e − t 2
2
for t ≥ 0 . Further, for t ≥ 2 ,
Ð ( ∥ x ∥ ≥ t + √ m ) ≤ e 1 − t 2
2 .
Proof. For the first claim, we only apply the change of variables theorem of integration.
For the second one, note that
δ ∥ x ∥ ( t + √ m ) = 1
2 m /2 − 1 Γ( m /2)
( t + √ m ) m − 1
2
e √ m ( t + √ m ) e m − t 2
2 .
Hence
Ð ( ∥ x ∥ ≥ t + √ m ) = ∫ ∞
t
δ ∥ x ∥ ( s + √ m ) d s
≤ ∫ ∞
t
1
2 m /2 − 1 Γ( m /2)
( s + √ m ) m − 1
2
e √ m ( s + √ m ) e m − s 2
2 d s
≤ e m − t 2
2 ∫ ∞
t
1
2 m /2 − 1 Γ( m /2)
( s + √ m ) m − 1
2
e √ m ( s + √ m ) d s
= e m − t 2
2 ∫ ∞
√ m ( t + √ m )
s m +1
2 − 1 e − s
2 m /2 − 1 m m +1
4 Γ( m /2)
d s
≤ e m − t 2
2 ∫ ∞
1
s m +1
2 − 1 e − s
2 m /2 − 1 m m +1
4 Γ( m /2)
d s
= e m − t 2
2
Γ( m +1
2 )
2 m /2 − 1 m m +1
4 Γ( m /2) ≤ e m
2 m
2 +1 m m − 1
4
e − t 2
2
where the last inequality follows from Γ ( m +1
2 ) /Γ( m
2 ) ≤ Γ ( m
2 + 1 ) /Γ( m
2 ) = m
2 . To finish the
proof, we just observe that e m
2 m
2 +1 m m − 1
4 ≤ e . □
1 §2 κ : a condition number for spherical algebraic sets
When can arbitrarily small perturbations of a polynomial tuple f ∈ H d [ q ] alter the local
topology of the zero set Z Ó ( f ) around a point x ∈ Ó n ? This can only happen at a singularity,
since all regular zeros look locally the same and all non-zeros also look the same. Motivated
by this, one defines the following local condition number.
Definition 1 §2 1 . Let f ∈ H d [ q ] and x ∈ Ó n , the local condition number of f at x , κ ( f , x ) ,
is the quantity in (0 , ∞ ] given by
κ ( f , x ) := ∥ f ∥ W
√ ∥ f ( x ) ∥ 2 + σ q (∆ − 1
d D x f ) 2
( 1.10 )
where σ q is the q th singular value, ∆ d as in ( 1.8 ) and D x f : T x Ó n → Ò q the tangent map.

34 Josué Tonelli-Cueto 1 §2
Since one expects the global topology not to change when the local topology does not
change at any point, this motivates the following definition.
Definition 1 §2 2 . Let f ∈ H d [ q ] , the global condition number of f , κ ( f ) , is the quantity in
(0 , ∞ ] given by
κ ( f ) := max
x ∈ Ó n κ ( f , x ) . ( 1.11 )
In the Definition 1 §2 1 , we note that, in the denominator of κ ( f , x ) , ∥ f ( x ) ∥ controls how
near is f of having a zero at x and σ q (∆ − 1
√ d D x f ) how near of not having full rank D x f is.
Thus κ ( f , x ) is large when f is near of having a singular zero at x . The following proposition
is a weak formalization of this observation. Recall that the zero set Z ( f ) := f − 1 (0) of a
map f : M → Ò q , where M is a smooth manifold, is called regular if for all x ∈ Z ( f ) ,
rank D x f = q .
Pr oposition 1 §2 1 . Let f ∈ H d [ q ] . Then Z Ó ( f ) := { x ∈ Ó n | f ( x )=0 } has a singularity
at x ∈ Ó n iff κ ( f , x ) = ∞ . In particular, Z Ó ( f ) is regular iff κ ( f ) < ∞ . □
In the next chapter, we will prove the following theorem which justifies the name condi-
tion number at least from the point of view of computing the homology groups.
Theor em 1 §2 2 . Let f ∈ H d [ q ] be such that κ ( f ) < ∞ . Then for all g ∈ B W ( f , κ ( f ) − 1 ∥ f ∥ W ) ,
H • ( Z Ó ( f ))  H • ( Z Ó ( g )) .
We will focus in the properties and possible bounds of the condition number. We will
finish with a discussion, showing that many possible alternative definitions are computation-
ally equivalent. Let us note that the definition of this condition number is the consequence of
a long sequence of works [143, 144, 133, 139, 140, 141, 142, 88, 91, 92] , so our choice of
properties is motivated by what this experience has shown to be important in the application
of κ as a condition number.
1 §2 -1 Fundamental properties
There are roughly four main properties of the local condition number that will be funda-
mental in our work. We go one by one.
Regularity inequality
The regularity inequality relates how near is x from being a zero of f with how near of
being singular D x f is. This will be used whenever we need to justify that we can compute
the pseudoinverse of D x f near the zero set of f . Recall that for a surjective linear map A,
the pseudoinverse , A † , is the linear map given by
A † := A ∗ ( AA ∗ ) − 1 ( 1.12 )
for which AA † = É and A † A is the orthogonal projection onto ( ker A ) ⊥ .
Pr oposition 1 §2 3 (Regularity inequality). [91 ; Proposition 3.6 ] . Let f ∈ H d [ q ] and x ∈
Ó n . Then either
∥ f ( x ) ∥
∥ f ∥ W ≥ 1
√ 2 κ ( f , x ) or σ q (∆ − 1
d D x f )
∥ f ∥ W ≥ 1
√ 2 κ ( f , x )
.

1 §2 Condition and Homology in Semialgebraic Geometry 35
In particular, if √ 2 κ ( f , x ) ∥ f ( x ) ∥
∥ f ∥ W < 1 , then D x f : T x Ó n → Ò q is surjective and its pseudoin-
verse D x f † exists.
Proof. Assume that neither of the alternatives holds, then
1
κ ( f , x ) = √ ( ∥ f ( x ) ∥
∥ f ∥ W ) 2
+ ( σ q (∆ d D x f )
∥ f ∥ W ) 2
< √ 1
2 κ ( f , x ) 2 + 1
2 κ ( f , x ) 2 = 1
κ ( f , x )
which gives a contradiction. Hence at least one of the options should hold. The last claims
follows from the fact that a matrix A ∈ Ò q × ( n +1) is surjective iff σ q ( A ) > 0 since q ≤ n . □
1st Lipschitz pr operty
The first Lipschitz property of the local condition number tells us that κ ( f , x ) − 1 ∥ f ∥ W as
a function of f is a Lipschitz function. This guarantees that sufficiently small errors in f do
not affect dramatically the condition number. Extending by continuity, we take κ (0 , x ) = ∞
and ∥ 0 ∥ W κ (0 , x ) − 1 = 0 .
Pr oposition 1 §2 4 (1st Lipschitz pr operty). Let x ∈ Ó n . Then the map
H d [ q ] → [0 , ∞ )
f 7→ ∥ f ∥ W
κ ( f , x )
is 1 -Lipschitz with respect to the Weyl norm.
Proof. By Definition 1 §2 1 and Proposition 1 §1 6 , we can write
∥ f ∥ W
κ ( f , x ) =   ( R 0
x ( f ) , σ q ( R 1
x ( f )) )  
where the right-hand side norm is the usual Euclidean norm. Then, by the triangle inequality,
    ∥ f ∥ W
κ ( f , x ) − ∥ ˜
f ∥ W
κ ( ˜
f , x )     ≤    ( R 0
x ( f ) − R 0
x ( ˜
f ) , σ q ( R 1
x ( f )) − σ q ( R 1
x ( ˜
f )) )    .
We know that σ q is 1 -Lipschitz with respect to the operator norm. Therefore
    ∥ f ∥ W
κ ( f , x ) − ∥ ˜
f ∥ W
κ ( ˜
f , x )     ≤    ( R 0
x ( f ) − R 0
x ( ˜
f ) , ∥ R 1
x ( f ) − R 1
x ( ˜
f ) ∥ )   
=    ( R 0
x ( f − ˜
f ) , ∥ R 1
x ( f − ˜
f ) ∥ )    ≤ ∥ R x ( f − ˜
f ) ∥ F ≤ ∥ f − ˜
f ∥ W = dist W ( f , ˜
f )
where, in the second line, the equality follows from the linearity of R x , the first inequality from
the inequality between the operator and the Frobenius norms, and the second inequality
from Proposition 1 §1 6 , which claims that R x is an orthogonal projection. The proposition is
proven. □
Cor ollary 1 §2 5 . The map
H d [ q ] → [0 , ∞ ]
f 7→ ∥ f ∥ W
κ ( f )
is 1 -Lipschitz with respect to the Weyl norm.

36 Josué Tonelli-Cueto 1 §2
Proof. Note that f 7→ ∥ f ∥ W / κ ( f ) is defined as the pointwise minimum of a family of non-
negative 1 -Lipschitz functions. Hence it is 1 -Lipschitz. □
Cor ollary 1 §2 6 . Let f ∈ H d [ q ] and x ∈ Ó n . Then κ ( f , x ) ≥ 1 and κ ( f ) ≥ 1 .
Proof. By applying Proposition 1 §2 4 to f and 0 , we have
∥ f ∥ W
κ ( f , x ) =     ∥ f ∥ W
κ ( f , x ) − ∥ 0 ∥ W
κ (0 , x )     ≤ ∥ f − 0 ∥ W = ∥ f ∥ W
which gives the desired claim. □
2nd Lipschitz pr operty
The second Lipschitz property of the local condition number establishes the Lipschitz-
ness with respect to the second argument.
Pr oposition 1 §2 7 (2nd Lipschitz pr operty). [88 ; Proposition 4.7 ] . Let f ∈ H d [ q ] . The
map
Ó n → [0 , 1]
x 7→ 1
κ ( f , x )
is D -Lipschitz with respect to the geodesic distance on Ó n .
To prove this property, we will use the following lemma.
Lemma 1 §2 8 . Let f ∈ H d [ q ] and A ∈ Ò ( n +1) × ( n +1) be an antisymmetric matrix. Then
D X f AX ∈ H d [ q ] and ∥ D X f AX ∥ ≤ D ∥ A ∥ ∥ f ∥ W .
Proof of Proposition 1 §2 7 . Let u be any orthogonal transformation taking x to ˜
x . Then we
have that κ ( f , ˜
x ) = κ ( f u , x ) . Therefore, by the 1st Lipschitz property,
    1
κ ( f , x ) − 1
κ ( f , ˜
x )     = 1
∥ f ∥ W     ∥ f ∥ W
κ ( f , x ) − ∥ f u ∥ W
κ ( f u , x )     ≤ ∥ f − f u ∥ W
∥ f ∥ W
where we have used that ∥ f ∥ = ∥ f u ∥ by Proposition 1 §1 4 .
Consider now the constant-speed path u : [0 , 1] → O ( n + 1) obtained by doing the
planar rotation between x and ˜
x from the zero angle to the full angle dist Ó ( x , ˜
x ) . By the
chain rule, d
d t f u ( t ) = D u ( t ) X f u ′ ( t ) X = D X ( f u ( t ) ) u ( t ) ∗ u ′ ( t ) X .
Therefore
∥ f − f u ∥ W =     ∫ 1
0
d
d t f u ( t ) d s     W ≤ ∫ 1
0   D X ( f u ( t ) ) u ( t ) ∗ u ′ ( t ) X   W d s .
Now, note that ∥ u ( t ) ∗ u ′ ( t ) ∥ = ∥ u ′ ( t ) ∥ = dist Ó ( x , ˜
x ) and that u ( t ) ∗ u ′ ( t ) is antisymmetric,
because u : [0 , 1] → O ( n + 1) is a planar rotation going from the angle zero to dist Ó ( x , ˜
x ) .
Hence, by Lemma 1 §2 8 , ∥ f − f u ∥ W ≤ D dist Ó ( x , ˜
x ) ∥ f ∥ W and the proof concludes. □

1 §2 Condition and Homology in Semialgebraic Geometry 37
Proof of Lemma 1 §2 8 . The present proof could be carried inside the framework of he reals,
but it would be too tedious to do so. 2 This is why we will work for this proof over the complex
numbers. Let u ∈ U ( n + 1) be the unitary transformation such that
u ∗ A u = ©   «
√ − 1 s 0
. . . √ − 1 s n ª ® ® ¬
=: √ − 1 S .
Then
( D X f AX ) u = D u X f A u X = D u X f u u ∗ A u X = D X f u ( √ − 1 S ) X
and so, by the complex version of Proposition 1 §1 4 , we can assume, without loss of gener-
ality, that A is already a purely imaginary diagonal matrix.
By direct computation,
( D X f ( √ − 1 S ) X ) i = ∑
| α | = d i
√ − 1 f i , α ( n
∑
k =0
α k s k ) X α = ∑
| α | = d i
√ − 1 f i , α ⟨ α , s ⟩ X α
where we use that both α and s are real in the last equality. Thus
∥ D X f ( √ − 1 S ) X ∥ 2
W = ∑
i , α ( d i
α ) − 1
| f i , α | 2 | ⟨ α , s ⟩ | 2 .
By Hölder’s inequality, | ⟨ α , s ⟩ | ≤ ∥ α ∥ 1 ∥ s ∥ ∞ ≤ D ∥ S ∥ , and the proof concludes. □
Condition number theor em
The local discriminant set at x is the set
Σ d [ q ] x := { g ∈ H d [ q ] | g ( x ) = 0 , rank D x g < q } ( 1.13 )
and the discriminant set the set
Σ d [ q ] := ∪ { Σ d [ q ] x | x ∈ Ó n } . ( 1.14 )
Once can see that Proposition 1 §2 1 can be reformulated as saying that
Σ d [ q ] x = { g ∈ H d [ q ] | κ ( g , x ) = ∞ } and Σ d [ q ] = { g ∈ H d [ q ] | κ ( g ) = ∞ } .
This claim can be made stronger by relating the condition number to the distance of f to
these sets. This is the so-called condition number theorem, which gives a nice geometric
interpretation of the condition number.
Theor em 1 §2 9 (Local condition number theor em). [87 ; Proposition 19.6 ] and [88 ; The-
orem 4.4 ] . Let f ∈ H d [ q ] and x ∈ Ó n . Then
κ ( f , x ) = ∥ f ∥ W
dist W ( f , Σ d [ q ] x )
where dist W is the distance with respect to the Weyl norm.
2 This is so, because over the complex numbers we can put A in diagonal form without loss of generality, but
this is not true over the reals, since its eigenvalues are either zero or imaginary.

38 Josué Tonelli-Cueto 1 §2
Proof. We only have to show that κ ( f , x ) ≤ ∥ f ∥ W
dist W ( f , Σ d [ q ] x ) , since the other inequality follows
directly from the 1st Lipschitz property (Proposition 1 §2 4 ) by letting the other polynomial
tuple to be in Σ d [ q ] x .
Let v ∈ ( ker ∆ − 1
d D x f ) ⊥ ∩ Ó n be such that
∥ ∆ − 1
d D x f v ∥ = σ q (∆ − 1
d D x f )
and ˜
f be the orthogonal projection of f onto { ev i
x , dev i
x , v | i ∈ [ q ] } ⊥ , i.e.,
˜
f := f − ∑
i ∈ [ q ] ⟨ f , ev i
x ⟩ ev i
x − ∑
i ∈ [ q ] ⟨ f , dev i
x , v ⟩ dev i
x , v
where, by Proposition 1 §1 1 , ⟨ f , ev i
x ⟩ = f i ( x ) and ⟨ f , dev i
x , v ⟩ is the i th component of
∆ − 1
d D x f v .
By the above, this means that ˜
f ( x ) = 0 and that
∆ − 1
d D x ˜
f = ∆ − 1
d D x f ( É − v v ∗ )
which has q th singular value equal to zero, because v was chosen to be the singular vector
associated to the q th singular value of ∆ − 1
d D x f . This means that κ ( ˜
f , x ) = ∞ and so
˜
f ∈ Σ d [ q ] x . Further, by Proposition 1 §1 1 ,
dist W ( f , ˜
f ) =       ∑
i ∈ [ q ] ⟨ f , ev i
x ⟩ ev i
x + ∑
i ∈ [ q ] ⟨ f , dev i
x , v ⟩ dev i
x , v       W
= √ ∥ f ( x ) ∥ 2 + ∥ ∆ − 1
d D x f v ∥ 2 = ∥ f ∥ W
κ ( f , x )
which implies that dist W ( f , Σ d [ q ] x ) ≤ ∥ f ∥ W / κ ( f , x ) , as desired. □
Cor ollary 1 §2 10 (Global condition number theor em). [87 ; Theorem 19.3 ] and [88 ; The-
orem 4.4 ] . Let f ∈ H d [ q ] . Then
κ ( f ) = ∥ f ∥ W
dist W ( f , Σ d [ q ])
where dist W is the distance with respect to the Weyl norm.
Proof. Just notice that dist W ( f , Σ d [ q ]) = min x ∈ Ó n dist W ( f , Σ d [ q ] x ) . □
In view of the above theorem, we can interpret Theorem 1 §2 2 as saying that no changes
in topology will occur as long as we don’t cross Σ d [ q ] .
Higher derivative estimate
The higher derivative estimate relates Smale’s γ with the condition number. Its useful-
ness relies on the fact that while for computing Smale’s γ one needs to evaluate all higher
derivatives of f , this is not necessary to evaluate the condition number. For f ∈ H d [ q ] and
x ∈ Ó n , let us define
µ ( f , x ) := ∥ f ∥ W
σ q (∆ − 1
d D x f ) = ∥ f ∥ W ∥ D x f † ∆ d ∥ ( 1.15 )

1 §2 Condition and Homology in Semialgebraic Geometry 39
where ∆ d is as in ( 1.8 ). We note that, in general, κ ( f , x ) ≤ µ ( f , x ) , with equality if f ( x ) = 0 ;
and that if √ 2 ∥ f ( x ) ∥
∥ f ∥ W < 1 , µ ( f , x ) ≤ √ 2 κ ( f , x ) , by the regularity inequality (Proposition 1 §2 3 ).
Recall that D x f refers to the tangent map T x Ó n → Ò q and D x f to the tangent map
T x Ò n +1 → Ò q . The notion of Smale’s gamma gives information about the magnitude of the
higher derivatives of an analytic function.
Definition 1 §2 3 . Let f : Ò m → Ò q be an analytic function and x ∈ Ò m be a point. Then
Smale’s gamma of f at x , γ ( f , x ) , is the non-negative real number given by
γ ( f , x ) := sup
k ≥ 2     1
k ! D x f † D k
x f    
1
k − 1 ( 1.16 )
where D k
x f is the tensor formed by the derivatives of order k of f , † is the pseudoinverse
and ∥ · ∥ is the operator norm. By convention, γ ( f , x ) = ∞ when D x f is not surjective.
Together with this notion, we introduce the notion of Smale’s projective γ , which is
Smale’s γ with the derivative substituted by the derivative on the sphere.
Definition 1 §2 4 . Let f ∈ H d [ q ] and x ∈ Ó n . Then Smale’s projective gamma of f at x ,
γ ( f , x ) , is the non-negative real number given by
γ ( f , x ) := sup
k ≥ 2     1
k ! D x f † D k
x f    
1
k − 1 ( 1.17 )
where D k
x f is the tensor formed by the derivatives of order k of f , † is the pseudoinverse
and ∥ · ∥ is the operator norm. By convention, γ ( f , x ) = ∞ when D x f is not surjective.
One can easily see that, in general, γ ( f , x ) ≤ γ ( f , x ) . The higher derivative estimate,
relates this quantity with µ .
Pr oposition 1 §2 11 (Higher derivative estimate). [87 ; Theorem 16.1 ] and [88 ; Proposi-
tion 4.1 ] . Let f ∈ H d [ q ] and x ∈ Ó n . Then
γ ( f , x ) ≤ γ ( f , x ) ≤ 1
2 D 3
2 µ ( f , x ) .
Proof. By Definition 1 §2 3 , it is enough to bound    1
k ! D x f † D k
x f    1
k − 1 for k ≥ 2 . Now,
    1
k ! D x f † D k
x f     ≤   D x f † ∆ d       1
k ! ∆ d D k
x f    
where    1
k ! ∆ d D k
x f    ≤ 1
D 1
2 ( D
k ) ∥ f ∥ W , by Corollary 1 §1 10 . Thus
    1
k ! D x f † D k
x f    
1
k − 1
≤ ( 1
D 1
2 ( D
k ) ) 1
k − 1
µ ( f , x ) 1
k − 1 .
Now, µ ( f , x ) ≥ κ ( f , x ) ≥ 1 , by Corollary 1 §2 6 , and so µ ( f , x ) 1
k − 1 ≤ µ ( f , x ) for k ≥ 2 .
Also, one can easily check that
sup
k ≥ 2 ( 1
D 1
2 ( D
k ) ) 1
k − 1
≤ 1
2 D 3
2 ,

40 Josué Tonelli-Cueto 1 §2
since for k ≥ 3 ,
( 1
D 1
2 ( D
k ) ) 1
k − 1
≤ ( D
k ) 1
k − 1
≤ ( D k
k ! ) 1
k − 1
= D 1+ 1
k − 1
( k !) 1
k − 1 ≤ D 3
2
( 2 k − 1 ) 1
k − 1
< 1
2 D 3
2 .
□
However, in our case, the following variant will be more useful since we will consider
sums of homogeneous polynomials and constants.
Cor ollary 1 §2 12 . [92 ; Proposition 4.5 ] . Let f ∈ H d [ q ] and x ∈ Ó n . Define
f Ó := ( f ,
n
∑
i =0
X 2
i − 1 ) . ( 1.18 )
Then
2 γ ( f Ó , x ) ≤ D 3
2 µ ( f , x ) + D 1
2 µ ( f , x ) ∥ f ( x ) ∥
∥ f ∥ W
+ 1 . ( 1.19 )
Proof. By direct computation,
D k
x f Ó ( u 1 , . . . , u k ) =
                                    
( D x f ( u 1 )
2 ⟨ x , u 1 ⟩ ) , if k = 1 ,
( D 2
x f ( u 1 , u 2 )
2 ⟨ u 1 , u 2 ⟩ ) , if k = 2 ,
( D k
x f ( u 1 , . . . , u k )
0 ) , if k > 2 .
Using this equality for k = 1 we deduce that ker D x f Ó = T x Ó n ∩ ker D x f = ker D x f . Let
V = ( ker D x f ) ⊥ ⊆ T x Ó n . Then ( ker D x f Ó ) ⊥ = V + Ò x and, for all λ ∈ Ò ,
D x f Ó ( v + λ x ) = ( D x f ( v ) + λ ∆ 2
d f ( x )
2 λ ) ( 1.20 )
where D x f ( x ) = ∆ 2
d f ( x ) follows from Euler’s identity for homogeneous functions.
By explicitly inverting the map in ( 1.20 ), we obtain
( D x f Ó ) † ( w
t ) = D x f † ( w − t
2 ∆ 2
d f ( x ) ) + t
2 x .
Thus
( D x f Ó ) † D k
x f Ó
k ! ( u 1 , . . . , u k ) =           
D x f † D 2
x f
2 ( u 1 , u 2 ) − ⟨ u 1 , u 2 ⟩
2 D x f † ∆ 2
d f ( x ) + ⟨ u 1 , u 2 ⟩
2 x , if k = 2 ,
D x f † D k
x f
k ! ( u 1 , . . . , u k ) , if k > 2 .

1 §2 Condition and Homology in Semialgebraic Geometry 41
Applying the triangle inequality and Definition 1 §2 3 , we obtain
γ ( f Ó , x ) ≤ γ ( f , x ) + 1
2 ∥ D x f † ∆ 2 f ( x ) ∥ + 1
2 ,
which implies
2 γ ( f Ó , x ) ≤ D 3
2 µ ( f , x ) + D 1
2 µ ( f , x ) ∥ f ( x ) ∥
∥ f ∥ W
+ 1
where the first term in the right-hand side follows from the higher derivative estimate (Propo-
sition 1 §2 11 ) and the second from the relations
∥ D x f † ∆ 2
d f ( x ) ∥ ≤ ∥ D x f † ∆ d ∥ ∥ ∆ d ∥ ∥ f ( x ) ∥ = ∥ f ∥ W ∥ D x f † ∆ d ∥ D 1
2 ∥ f ( x ) ∥
∥ f ∥ W
and the definition of µ . This finishes the proof. □
1 §2 -2 Bounds: worst-case and probabilistic
We present two kinds of bounds for the condition number, which try to go around the
fact that the possible maximum of κ is infinite. The first one puts a restriction on the f ∈ H d [ q ]
we consider, like restricting the coefficients to be integers of at most a certain size, and looks
for a bound that holds for all f satisfying the restriction for which κ ( f ) is finite. The second
one considers a random f ∈ H d [ q ] , such as a KSS random polynomial tuple, and looks for
a tail bound for the random variable κ ( f ) . Each of them represents a different philosophy in
the complexity of numerical algorithms.
Gap theor em for integer inputs
The idea of bounding the condition number in terms of the bit size goes back to Rene-
gar [331] . The underlying philosophy is to translate the condition-based estimates of the
condition-using complexity theorist to something that the classical computer scientist can
understand, like a worst-case bit complexity estimate. In linear programming, the bound
by Renegar (see [87 ; Proposition 7.9 ] and associated remarks) was successful in providing
bounds giving the desired complexity for a series of algorithms using condition numbers.
However, in our case, the bounds obtained will not allow us to get good bit complexity
estimates for the algorithms under study.
Theor em 1 §2 13 . Let f ∈ H d [ q ] be such that all its coefficients are integers of absolute
value at most H . Then either κ ( f ) = ∞ or
κ ( f ) ≤ √ 2 DNH ( 2 2 − 1+ n + q
4 D 3+ n + q
2 H √ N ) (1+ n + q )(4 D ) 1+ n + q
= O ( DHN ) O ( D ) 1+ n + q
.
Proof. The proof relies on a generalization of Polya’s theorem by Jeronimo, Perruci and
Tsigaridas [236 ; Theorem 1 ] . To apply their theorem to our case, we note that
2 D ( ∥ f ∥ W
κ ( f ) ) 2
≥ min
x ∈ Ó n ( ∥ f ( x ) ∥ 2 + σ q (∆ − 1
d D x f ) 2 ) .
To obtain the minimum in the right-hand side as a minimum of a polynomial function, consider
the map g : ( x , v ) 7→ ∥ f ( x ) ∥ 2 + ∥ v ∗ D x f ∥ 2 and minimize it in the compact semialgebraic
set C given by
n
∑
i =0
x 2
i = 1 ,
q
∑
i =1
d i v 2
i = 1 .

42 Josué Tonelli-Cueto 1 §2
Then
2 D ( ∥ f ∥ W
κ ( f ) ) 2
≥ min
( x , v ) ∈ C g ( x , v ) ,
since σ q ( D x f ) = min { ∥ D X f v | ∥ v ∥ = 1 , x ⊥ v } . Now, a direct computation and a rough
estimation shows that g is a polynomial of degree 2 D whose coefficients are integers of
absolute value at most D 2 H 2 N. Applying [236 ; Theorem 1 ] , we obtain
min
( x , v ) ∈ C g ≥ ( 2 4 − 1+ n + q
2 ( D 2 H 2 N )(2 D ) 1+ n + q ) − (1+ n + q )2 1+ n + q (2 D ) 1+ n + q
and so the result follows after minor computations and estimations, since for the given f ,
we have ∥ f ∥ W ≤ H √ N. □
Remark 1 §2 1 . The above bound is novel and it was a missing ingredient in the existing theory,
when compared to the condition-based complexity framework applied to other problems.
It clearly does not lead to single exponential bounds of κ , which controls the run time of
the algorithm; although it can be used to guarantee that log κ , which controls the precision
needed by the algorithm, is singly exponential in n , polynomial in the degree D and linear in
the bit size log H of the coefficients of f . ¶
Remark 1 §2 2 . [236 ; Theorem 1 ] is a very general result that we apply it to a very particular
setting. It would be interesting to see, if using the techniques in [236] and in [235] , one can
provide a better bound such as the one in the question below.
Open problem A . Let f ∈ H d [ q ] be such that all its coefficients are integers of absolute
value at most H . Is it true that either κ ( f ) = ∞ or
κ ( f ) ≤ O ( DHN ) 2 O ( n ) ?
A bound like the above would give the grid method the same worst case complexity as
CAD, for integer inputs. Also, it would make the precision to be linear in the logarithm of the
degree, instead of polynomial in the degree. ¶
Pr obabilistic bounds
The problem of computing a bound for κ ( f ) for restricted f ∈ H d [ q ] is that a bad f can
spoil the full basket. In the same way that we don’t judge a community by its worst members,
we should not just judge a parameter by its worst value. Behind this way of thinking, trying to
understand the full statistics of a behaviour and not just the worst behaviour, lies the founding
idea of considering probabilistic bounds of κ . We give two ways of arriving to a probabilistic
bound: via integral geometry and via geometric analysis.
We observe that there are two frameworks to obtain probabilistic bounds: the average
and the smoothed framework. The difference between them relies on the randomness model
used to obtain tail bounds of the condition number. In the average framework, introduced
by Goldstine and von Neumann [194] , Demmel [149] and Smale [374] , one considers a
random polynomial tuple f ∈ H d [ q ] which has a normal or uniform distribution. This random
input is suppose to represent the “average” input that one will find and so the usual behaviour
that one will find. In the smoothed framework, introduced by Spielman and Teng [380] , one
considers a random polynomial tuple f σ ∈ H d [ q ] of the form f σ := f + σ g with f ∈ H d [ q ]

1 §2 Condition and Homology in Semialgebraic Geometry 43
fixed, σ > 0 and g with a normal or uniform distribution. The random f σ represents an input
f with some random perturbation, whose magnitude is controlled by σ . In this way, one
hopes to get a bound on the worst probabilistic behaviour of a randomly perturbed input.
One should note that as σ grows, one recovers the average framework.
V ia integral geometry The integral geometric approach relies heavily on the condi-
tion number theorem (Corollary 1 §2 10 ) and the geometry of the ill-posed set.
Theor em 1 §2 14 . [87 ; Theorem 21.1 ] . Let Σ ⊆ Ò N be a set and C : Ò N \ { 0 } → [1 , ∞ ] be
given by
C ( x ) := ∥ x ∥
dist ( x , Σ) .
Assume that there is a homogenous polynomial of degree d containing Σ in its zero set.
(A) Let X ∈ Ò N be a Gaussian random vector. Then for t ≥ (2 d + 1)( N − 1) ,
Ð ( C ( X ) ≥ t ) ≤ 11 d ( N − 1) 1
t
and
Å log C ( X ) ≤ log ( N − 1) + log d + log (30) .
(S) Let x ∈ Ó N , σ ∈ [0 , 1] and X σ ∈ Ò N a random vector uniformly distributed in B ( x , σ ) .
Then for t ≥ (2 d + 1)( N − 1) σ − 1 ,
Ð ( C ( X σ ) ≥ t ) ≤ 11 d ( N − 1) σ − 1 1
t
and
Å log C ( X σ ) ≤ log ( N − 1) + log d + log σ − 1 + log (30) . □
In order to apply this theorem, we need to prove that Σ d [ q ] is contained in some hyper-
surface. This can be easily done, using techniques from algebraic geometry. However, we
omit the proof as these techniques go beyond the scope of this thesis.
Pr oposition 1 §2 15 . [88 ; Proposition 4.20 ] . There is an integer polynomial of degree at most
n 2 n D n such that Σ d [ q ] is contained in its zero set.
Proof. The claim is true for q ≤ n + 1 by [88 ; Proposition 4.20 ] . To extend it further, recall that
for q ≥ n + 1 , κ ( f , x ) = ∥ f ∥ / ∥ f ( x ) ∥ . Therefore the linear projection H d [ q ] → H d [ n + 1]
maps surjectively Σ d [ q ] onto Σ d [ n +1] for q ≥ n +1 and the claim holds also for q ≥ n +1 . □
Combining the above two results, we obtain the following probabilistic bound on κ .
Pr oposition 1 §2 16 . (A) Let f ∈ H d [ q ] be a KSS random polynomial tuple. Then for t ≥
( n 2 n +1 D n + 1)( N − 1) ,
Ð ( κ ( f ) ≥ t ) ≤ 11( n 2 n +1 D n + 1)( N − 1) 1
t
and
Å log κ ( f ) ≤ log ( N − 1) + n ( log D + 3 log 2) + log (30) .

44 Josué Tonelli-Cueto 1 §2
(S) Let f ∈ H d [ q ] , σ ∈ [0 , 1] and f σ ∈ H d [ q ] a random polynomial tuple uniformly dis-
tributed in B W ( f , σ ) . Then for t ≥ ( n 2 n +1 D n + 1)( N − 1) σ − 1 ,
Ð ( κ ( f σ ) ≥ t ) ≤ 11( n 2 n +1 D n + 1)( N − 1) σ − 1 1
t
and
Å log κ ( f σ ) ≤ log ( N − 1) + n ( log D + 3 log 2) + log σ − 1 + log (30) . □
Remark 1 §2 3 . The above results do not apply to the local condition number. The main rea-
son is that the above theorem, coming from [90] of Bürgisser, Cucker and Lotz, only covers
the case in which Σ is contained in an algebraic hypersurface, but it does not take advantage
of the fact that Σ might have higher codimension. An extension of this form was obtained
by Lotz [281] , but one should still work the details carefully as in the statement of Theo-
rem 1 §2 14 .
Open problem B . Can the bound in Theorem 1 §2 14 be extended to the case in which Σ is
a real algebraic variety of degree d and codimension c in such a way that the probability tail
bounds are of the form O ( d ( N − 1) t − c ) ?
Also, the following problem might be useful given the multihomogeneous structure of
Σ d [ q ] .
Open problem C . Can the bound in Theorem 1 §2 14 be improved in the case in which
Σ is a real algebraic hypersurface in Ò ∑ q
i =1 N i given by a multihomogeneous polynomial
h ( X 1 , . . . , X q ) of degree d i with respect to the block of N i variables X i ? More concretely,
let x 1 , . . . , x q be random vectors uniformly distributed in the unit balls. Is it true that
Ð ( C ( x ) ≥ t ) ≤ O ( √ q max
i ∈ [ q ] ( d i ( N i − 1)) 1
t )
as one obtains in the reducible case Σ = ∪ i ∈ [ q ] Σ i ? ¶
Remark 1 §2 4 . One should notice that N − 1 does not appear dividing in the above bounds
and successive bounds as it appears in the bounds given in [91, 92] . This is the case,
because there was a mistake in the citation of [87 ; Theorem 21.1 ] in [91] . However, this
mistake does not affect the order of the estimates. ¶
V ia geometric functional analysis The geometric functional analysis framework in
the probabilistic analysis of κ is quite new. It was introduced by Ergür, Paouris and Ro-
jas [175, 176] for the zero dimensional case and it was applied by Cucker, Ergür and the
author [136] to the case of a single polynomial. The advantage of this method is that it can
be applied to distributions more general than the normal distribution. We will show this in
Chapter 5 in the special case of hypersurfaces. Here, we will provide a tail bound with a very
simple proof for the Gaussian case.
Theor em 1 §2 17 . (A) Let f ∈ H d [ q ] be a KSS random polynomial tuple and x ∈ Ó n . Then
for t ≥ 2 ,
Ð ( κ ( f , x ) ≥ t ) ≤ 7
2 ( 119 N
n + 1 ) n +1
2 ( ln 1
2 t
t ) n +1
.

1 §2 Condition and Homology in Semialgebraic Geometry 45
(S) Let f ∈ H d [ q ] , σ > 0 , f σ := f + σ ∥ f ∥ W g be a random polynomial tuple such that
g ∈ H d [ q ] is a KSS random polynomial tuple and x ∈ Ó n . Then for t ≥ 2 ,
Ð ( κ ( f σ , x ) ≥ t ) ≤ 7
2 ( 119 N
n + 1 ) n +1
2 ( ln 1
2 t
t ) n +1 ( 1 + 1
σ ) n +1
.
In the proof, we will use the following proposition. Recall that ω m := π m
2 /Γ ( m
2 + 1 ) is
the volume of the unit Euclidean m -ball. We also recall Stirling’s estimation of Euler’s Gamma
function:
(2 e ) x
2
√ π x x +1
2
e − 1
6 x ≤ 1
Γ ( x
2 + 1 ) ≤ (2 e ) x
2
√ π x x +1
2
( 1.21 )
for x > 0 .
Pr oposition 1 §2 18 . (V) Let x ∈ Ò q , σ > 0 and x ∼ N ( x , σ ) be a random vector. Then
for all ε > 0 ,
Ð x ( ∥ x ∥ ≤ ε ) ≤ ω q
(2 π ) q
2 ( ε
σ ) q
.
(M) Let q ≤ n , A ∈ Ò q × n , σ > 0 and A ∼ N ( A , σ ) be a random matrix. Then for all ε > 0 ,
Ð A ( σ q ( A ) ≤ ε ) ≤ √ q (2 π ) q
2
2 ( e
2 π ) n
2 ω n +1 − q ( ε
σ ) n +1 − q
.
Proof of Proposition 1 §2 18 . (V) can be found at the end of [87 ; Proof of Proposition 4.21 ] ,
and (M) in [87 ; Proof of Proposition 4.19 ] (page 90), where here the factor ( e
1 − λ ) (1 − λ ) n
2 , with
λ = q − 1
n , is bounded by e n
2 . □
Proof of Theorem 1 §2 17 . (A) We do the proof for q < n + 1 , for q ≥ n + 1 is analogous. 3
By Proposition 1 §1 11 , we can see that
Ð ( ∥ f ∥ W ≥ t ) ≤ e 1 − t 2
8
for t ≥ 2 √ N. Now, note that for all t > 0 and u ≥ 2 √ N,
Ð ( κ ( f , x ))
= Ð ( ∥ f ∥ W / √ ∥ R 0
x ( f ) ∥ 2 + σ q ( R 1
x ( f )) ≥ t ) (By Proposition 1 §1 6 )
≤ Ð ( ∥ f ∥ W ≥ u or √ ∥ R 0
x ( f ) ∥ 2 + σ q ( R 1
x ( f )) 2 ≤ u / t ) (Implication bound)
≤ Ð ( ∥ f ∥ W ≥ u ) + Ð ( √ ∥ R 0
x ( f ) ∥ 2 + σ q ( R 1
x ( f )) 2 ≤ u / t ) (Union bound)
≤ Ð ( ∥ f ∥ W ≥ u ) + Ð ( ∥ R 0
x ( f ) ∥ ≤ u / t and σ q ( R 1
x ( f )) ≤ u / t ) (Implication bound)
≤ Ð ( ∥ f ∥ W ≥ u ) + Ð ( ∥ R 0
x ( f ) ∥ ≤ u / t ) Ð ( σ q ( R 1
x ( f )) ≤ u / t ) ( R 0
x ( f ) , R 1
x ( f ) independent)
≤ e 1 − u 2
8 + √ q
2 ω q ω n +1 − q ( e
2 π ) n
2 ( u
t ) n +1 (Proposition 1 §2 18 ) .
3 And for q > n + 1 , one can get a tail bound of the order O ( ln n +1
2 t − q ) , but this bound will not be useful for
us.

46 Josué Tonelli-Cueto 1 §2
We can apply Proposition 1 §2 18 , because, by Proposition 1 §1 6 , R 0
x ( f ) and R 1
x ( f ) come from
taking independent orthogonal projections of f . This fact guarantees that R 0
x ( f ) and R 1
x ( f )
are independent and Gaussian.
Here, we substitute u = 2 √ 2 N ln 1
2 t ≥ 2 √ N for t ≥ 2 , and we get for t ≥ 2 ,
Ð ( κ ( f , x )) ≤ ( e + √ q
2 ω q ω n +1 − q ( e
2 π ) n
2 (8 N ) n +1
2 ) ( ln 1
2 t
t ) n +1
since t − N ≤ ( ln 1
2 t
t ) n +1 . We substitute now the formula for ω m and we use the Stirling
estimation ( 1.21 ), to obtain the bound
√ q
2 ω q ω n +1 − q ( e
2 π ) n
2 ≤ √ e π
2( n − q + 1)
e n
( n − q + 1) n − q +1
2 q q
2
≤ √ e π
2
e n
( n − q + 1) n − q +1
2 q q
2
.
We put λ = q
n +1 ∈ [0 , 1] and observe that
( n − q + 1) n − q +1
2 q q
2 = ( (1 − λ ) 1 − λ
2 λ λ
2 ) n +1
( n + 1) n +1
2 ≥ ( n + 1
2 ) n +1
2
.
This gives us
e n
( n − q + 1) n − q +1
2 q q
2 ≤ e n
( n +1
2 ) n +1
2 ≤ e n ( 1 − 1
2 ln n +1
2 ) = 1
e ( 2 e 2
n + 1 ) n +1
2
≤ 1
e ( 119/8
n + 1 ) n +1
2
and so the desired bound follows using that e + √ π
2 e < 7/2 and that ( 119 N
n +1 ) n +1
2 ≥ 1 .
(S) For the smoothed case, the proof is analogous to the one above. In this case, we
have
Ð ( ∥ f σ ∥ W ≥ t ∥ f ∥ W ) ≤ Ð ( ∥ g ∥ W ≥ ( t − 1)/ σ ) ≤ e 1 − ( t − 1) 2
8 σ 2
for t ≥ 1+2 σ √ N. We proceed as above, but we use the general version of Proposition 1 §2 18
and we substitute u = ∥ f ∥ (2 √ 2 N σ ln 1
2 t + 1) . To obtain the same line or arguments as
above, one should only notice that u ≤ ∥ f ∥ 2 √ 2 N ln 1
2 t ( σ + 1) and so one obtains the same
as above, but with the extra factor ( 1 + 1
σ ) n +1 . □
To pass from the local condition estimates to the global ones, we use a grid-based
method which is common in the probabilistic methods coming from geometric functional
analysis (see [399 ; §4.4 ] ).
Theor em 1 §2 19 . (A) Let f ∈ H d [ q ] be a KSS random polynomial tuple. Then for t ≥ 4 ,
Ð ( κ ( f ) ≥ t ) ≤ 21 ( 1984 N
n + 1 ) n +1
2 D n ln n +1
2 t
t .

1 §2 Condition and Homology in Semialgebraic Geometry 47
(S) Let f ∈ H d [ q ] , σ > 0 , f σ := f + σ ∥ f ∥ g be a random polynomial tuple such that
g ∈ H d [ q ] is a KSS random polynomial tuple. Then for t ≥ 4 ,
Ð ( κ ( f σ ) ≥ t ) ≤ 21 ( 1984 N
n + 1 ) n +1
2 D n ln n +1
2 t
t ( 1 + 1
σ ) n +1
.
For proving the above theorem, we need the following lemma for “constructing” an
optimal grid that allows us to sample efficiently in the sphere and pass from the tail bound
estimates of the probability of the local condition number to the ones of the global condition
number.
Lemma 1 §2 20 . [398 ; Lemma 5.2 ] (cf. [399 ; §4.2 ] ). Let δ ∈ (0 , π /3) . Then there is a finite
set N δ ⊆ Ó n such that for all y ∈ Ó n , dist Ó ( y , N δ ) < δ and such that
# N δ ≤ 3 ( 3
δ ) n
.
Proof of Theorem 1 §2 19 . (A) Fix t ≥ 4 . Let G ⊆ Ó n be such that for all y ∈ Ó n , dist Ó ( y , G ) <
1
D t and define
κ G ( f ) := max
x ∈ G κ ( f , x ) .
By the 2nd Lipschitz property (Proposition 1 §2 7 ),
1
κ G ( f ) − 1
κ ( f ) ≤ D 1
D t = 1
t
and so
κ G ( f ) ≥ κ ( f )
1 + κ ( f )
t ≥ t
2 .
Hence κ ( f ) ≥ t implies κ G ( f ) ≥ t /2 and so, by the implication bound,
Ð ( κ ( f ) ≥ t ) ≤ Ð ( κ G ( f ) ≥ t /2) ≤ # G max
x ∈ G Ð ( κ ( f , x ) ≥ t /2)
where the last inequality is the union bound. Without loss of generality, we can assume
# G ≤ 3(3 D t ) n , by Lemma 1 §2 20 (taking δ = 1/( D t ) ). Finally, Theorem 1 §2 17 together
with some computations finishes the proof.
(S) Exactly like (A). □
Proof of Lemma 1 §2 20 . Let N δ be a maximal set of Ó n such that for each distinct x , ˜
x ∈
N δ , dist Ó ( x , ˜
x ) ≥ δ . Such a set exists and its finite, since otherwise we can construct a
sequence { x k } in Ó n such that for all distinct k , l ∈ Î , dist Ó ( x k , x l ) ≥ δ contradicting the
compactness of Ó n . This set satisfies the first property, since for all y ∈ Ó n , dist ( y , N δ ) < δ ,
otherwise N δ ∪ { y } would contradict the maximality of N δ .
By construction, the family of sets { B Ó ( x , δ /2) | x ∈ N δ } is disjoint and so
# N δ vol n ( B Ó ( e 0 , δ /2)) = vol n ∪ { B Ó ( x , δ /2) | x ∈ N δ } ≤ vol n ( Ó n ) = ( n + 1) ω n +1 .
Now, by [87 ; Lemma 2.31 ] ,
vol n ( B Ó ( e 0 , δ /2)) = n ω n ∫ δ
2
0
sin n − 1 s d s ≥ ω n ∫ δ
2
0
n sin n − 1 s cos s d s = ω n sin n δ
2 .

48 Josué Tonelli-Cueto 1 §2
Note that sin x ≥ 3 x
π for x ∈ (0 , π /6) , so sin n δ
2 ≥ ( 3 δ
2 π ) n . Finally, a simple maximization
shows that the bound provided is correct. □
Remark 1 §2 5 . The geometric functional analysis’ approach definitely provides a more ele-
mentary proof than the integral geometric approach. This can be a good pedagogical tool
when one cannot go into the details of the integral geometric proof. However, the bounds
obtained are worse, both in the constants and asymptotically. The latter seems to be an
intrinsic characteristic of the method (since similar issues happen in [175, 176] ), but it may
be because some of the traditionally used probabilistic bounds 4 are not the best bounds for
the job.
Open problem D . Can one obtain tail bounds of the form
Ð ( κ ( f , x ) ≥ t ) ≤ O ( poly ( N ) poly ( n ) 1
t n +1 ) and Ð ( κ ( f ) ≥ t ) ≤ O ( poly ( D , N ) poly ( n ) 1
t )
using the methods of geometric functional analysis? For an approach looking to prove the
above estimate, the techniques from [141] might be useful.
Despite all the above said about the drawbacks of the methods from geometric func-
tional analysis, we have to point out that these methods have an advantage over the methods
from integral geometry: they are able to handle probability distributions that are not normal.
This was shown for the zero dimensional case in [175, 176] and for the case of a single
polynomial in [136] , which we will cover in Chapter 5 . ¶
1 §2 -3 Alternative definitions of κ
Now that we have gone through all the theory, one might wonder about alternative
definitions of κ that might be better from one perspective or another. The following proposi-
tion however shows that all natural variations are equivalent up to a reasonable constant or
parameter.
Pr oposition 1 §2 21 . Let f ∈ H d [ q ] and x ∈ Ó n . Then:
(0) For κ 0 ( f , x ) := ∥ f ∥ W / max { ∥ f ( x ) ∥ , σ q (∆ − 1
d D x f ) } ,
1
√ 2 κ 0 ( f , x ) ≤ κ ( f , x ) ≤ κ 0 ( f , x ) .
(1) For κ 1 ( f , x ) := ∥ f ∥ W / √ ∥ f ( x ) ∥ 2 + σ q ( D x f ) 2 ,
κ 1 ( f , x ) ≤ κ ( f , x ) ≤ √ D κ 1 ( f , x ) .
(2) For κ 2 ( f , x ) := ∥ f ∥ W / √ ∥ f ( x ) ∥ 2 + σ q (∆ − 1
d D x f ) 2 , where D x f is the derivative of f as
a map on Ò n +1 ,
κ 2 ( f , x ) ≤ κ ( f , x ) ≤ √ 1 + D κ 2 ( f , x ) .
4 Specially Ð ( x / y ≥ t ) ≤ Ð ( x ≥ u or y ≤ u t ) ≤ Ð ( x ≥ u ) + Ð ( y ≤ u t ) .

1 §3 Condition and Homology in Semialgebraic Geometry 49
(3) For κ 3 ( f , x ) := ∥ ˆ
f ∥ W / √ ∥ ˆ
f ( x ) ∥ 2 + σ q (∆ − 1
d D x ˆ
f ) 2 where ˆ
f := ( f i / ∥ f i ∥ W ) i ∈ [ q ] ,
1
√ q κ 3 ( f , x ) ≤ ∥ f ∥ W
√ q max i ∈ [ q ] ∥ f i ∥ W
κ 3 ( f , x ) ≤ κ ( f , x ) ≤ ∥ f ∥ W
√ q min i ∈ [ q ] ∥ f i ∥ W
κ 3 ( f , x ) .
Proof. (0) follows from the inequality between the ℓ ∞ and ℓ 2 -norms in Ò 2 , (1) from
1
√ D σ q ( D x f ) ≤ σ q (∆ − 1
d D x f ) ≤ σ q ( D x f ) ,
(2) from
σ q (∆ − 1
d D x f ) ≤ σ q (∆ − 1
d D x f ) = σ q (∆ − 1
d D x f ( É − x x ∗ )+∆ d f ( x ) x ∗ )
≤ √ σ q (∆ − 1
d D x f ) 2 + D ∥ f ( x ) ∥ 2 ,
and (3) from
∥ f ∥ W
max i ∈ [ q ] ∥ f i ∥ W ∥ f ( x ) ∥
∥ f ∥ W ≤ ∥ ˆ
f ( x ) ∥ ≤ ∥ f ∥ W
min i ∈ [ q ] ∥ f i ∥ W ∥ f ( x ) ∥
∥ f ∥ W
and
∥ f ∥ W
max i ∈ [ q ] ∥ f i ∥ W
σ q (∆ − 1
d D x f )
∥ f ∥ W ≤ σ q (∆ − 1
d D x ˆ
f ) ≤ ∥ f ∥ W
min i ∈ [ q ] ∥ f i ∥ W
σ q (∆ − 1
d D x f )
∥ f ∥ W
.
□
The above proposition only shows some of the variations, which can be themselves
combined. In the end, the reason to choose the definition that we have chosen is conceptual
and historical, since it is the one that allows us to develop the theory in its maximum aesthetic
appealing.
Remark 1 §2 6 . Among the variants of κ above, the only one that is not strongly equivalent
is κ 3 . This is so, because κ 3 is small whenever κ is so, but the opposite impliciation is not
true. To see this, note that κ 3 is invariant under scalar multiplication of each component of
f , while this property is not true for κ . By taking one component to zero, we can construct
a sequence { f k } such that { κ 3 ( f k ) } is constant, but { κ 3 ( f k ) } goes to infinity. ¶
Remark 1 §2 7 . One can observe that Proposition 1 §2 21 (2) is just [142 ; Proposition 6.1 ] .
However, the difference in the constants is due to an error in the proof in [142] , in which
they used the identity f i ( e 0 ) = ∂ f / ∂ X 0 ( e 0 ) instead of the correct f i ( e 0 ) = d i ∂ f / ∂ X 0 ( e 0 ) ,
due to Euler’s identity. ¶
1 §3 κ : a condition number for spherical semialgebraic sets
A spherical semialgebraic set is a semialgebraic subset of the sphere described by
homogeneous polynomial. Given f ∈ H d [ q ] and Φ a Boolean formula over f , the spherical
semialgebraic set described by ( f , Φ) , S ( f , Φ) , is given by
S ( f , Φ) := Ó n ∩ W ( f , Φ) ( 1.22 )

50 Josué Tonelli-Cueto 1 §3
where W ( p , Φ) was defined in ( 0.2 ). Instead of defining a condition number for each descrip-
tion ( f , Φ) , we will define a condition number associated only to f . This condition number
will be finite when for all Φ , the descriptions ( f , Φ) is well-posed. The way to achieve this is
to focus on the possible boundary pieces. This motivates the following definition.
Definition 1 §3 1 . Let f ∈ H d [ q ] and x ∈ Ó n . The local intersection condition number of f
at x , κ ( f , x ) , is the quantity in [1 , ∞ ] given by
κ ( f , x ) := max
L ∈ [ q ] ≤ n +1 κ ( f L , x ) ( 1.23 )
where [ q ] ≤ n +1 := { K ⊆ [ q ] | # K ≤ n + 1 } and f L = ( f i ) i ∈ L , and the global intersection
condition number of f , κ ( f ) , the quantity given by
κ ( f ) := max
L ⊆ [ q ]
# L ≤ n +1
κ ( f L ) = max
y ∈ Ó n κ ( f , y ) . ( 1.24 )
The first thing we should notice is the following proposition, which explains the term
“intersection” in the name of the above condition number. Recall that an intersection ∩ i ∈ I N
of a family of smooth submanifolds { N i } i ∈ I of a smooth manifold M is called transversal in
M if for all x ∈ ∩ i ∈ I N i , ∑
i ∈ I
codim T x M T x N i = codim T x M ∩
i ∈ I
T x N i . ( 1.25 )
Pr oposition 1 §3 1 . Let f ∈ H d [ q ] . Then κ ( f ) < ∞ iff the following conditions hold:
(1) For all i ∈ [ q ] , Z Ó ( f i ) := S ( f , f i = 0) is regular.
(2) For all L ⊆ [ q ] , the intersection ∩ i ∈ I Z Ó ( f i ) is transversal in Ó n . □
Remark 1 §3 1 . One should observe that in the definition of κ , we limit to subsets L ⊆ [ q ] of
size at most n + 1 . However, this is not the case in Proposition 1 §3 1 . The reason for this is
that for # L ≥ n + 1 , ∩ i ∈ L Z Ó ( f i ) is transversal in Ó n iff ∩ i ∈ L Z Ó ( f i ) = ∅ . Therefore when
# L ≥ n + 1 , ∩ i ∈ L Z Ó ( f i ) is transversal in Ó n iff for all H ⊆ L with # H = n + 1 , ∩ i ∈ H Z Ó ( f i )
is transversal in Ó n .
We have in this case a theorem even more general than Theorem 1 §2 2 , which involves
not only algebraic sets, but all possible semialgebraic sets that can be constructed from f .
The proof will be done in the next chapter, together with the one of Theorem 1 §2 2 .
Theor em 1 §3 2 . Let f ∈ H d [ q ] be such that κ ( f ) < ∞ . Then for all Boolean formula Φ
over f and every g ∈ B W ( f , κ ( f ) − 1 min i ∥ f i ∥ W ) , we have H • ( S ( f , Φ))  H • ( S ( g , Φ)) .
As with Theorem 1 §2 2 , we can interpret Theorem 1 §3 2 as saying that homology will not
change as long as we do not cross the set of ill-posed polynomial tuples. In this case, we
have that the set of ill-posed sets is given by
Σ d [ q ] x := ∪
L ∈ [ q ] ≤ n +1
Σ L
d [ q ] x and Σ d [ q ] := ∪
L ∈ [ q ] ≤ n +1
Σ L
d [ q ] = ∪
z ∈ Ó n
Σ d [ q ] z ( 1.26 )

1 §3 Condition and Homology in Semialgebraic Geometry 51
where
Σ L
d [ q ] x := { g ∈ H d [ q ] | g L ( x ) = 0 , rank D x g L < q } and Σ L
d [ q ] := ∪
z ∈ Ó n
Σ L
d [ q ] z . ( 1.27 )
Transferring the properties from κ to κ is straightforward, with the exception of the condi-
tion number theorem (Theorem 1 §2 9 and Corollary 1 §2 10 ) that can only be transferred in a
weaker sense.
Pr oposition 1 §3 3 . Let f ∈ H d [ q ] and x ∈ Ó n . Then:
• Regularity inequality : For all L ∈ [ q ] ≤ n +1 , either
∥ f L ( x ) ∥
∥ f L ∥ W ≥ 1
√ 2 κ ( f , x ) or σ q (∆ − 1
d D x f L )
∥ f L ∥ W ≥ 1
√ 2 κ ( f , x )
.
In particular, for all L ∈ [ q ] ≤ n +1 , if √ 2 κ ( f , x ) ∥ f L ( x ) ∥
∥ f L ∥ W < 1 , then D x f L : T x Ó n → Ò L is
surjective and its pseudoinverse ( D x f L ) † exists.
• 1st Lipschitz property : The maps
H d [ q ] → [0 , ∞ )
g 7→ ∥ g ∥ W
κ ( g , x )
and H d [ q ] → [0 , ∞ )
g 7→ ∥ g ∥ W
κ ( g )
are 1 -Lipschitz with respect to the Weyl norm. In particular, κ ( f , x ) ≥ 1 and κ ( f ) ≥ 1 .
• 2nd Lipschitz property : The map
Ó n → [0 , 1]
y 7→ 1
κ ( f , y )
is D -Lipschitz with respect to the geodesic distance on Ó n .
• Weak condition number theorem :
κ ( f , x ) ≤ ∥ f ∥ W
dist W ( f , Σ d [ q ] x ) and κ ( f ) ≤ ∥ f ∥ W
dist W ( f , Σ d [ q ])
where dist W is the distance induced by the Weyl norm.
Proof. They follow straightforwardly from the properties of κ and the definition of κ . Ex-
panding this out, the regularity inequality follows from Proposition 1 §2 3 , the 1st Lipschitz
property from Proposition 1 §2 4 and Corollaries 1 §2 5 and 1 §2 6 , the 2nd Lipschitz property
from Proposition 1 §2 7 , and the weak condition number theorem from Theorem 1 §2 9 and
Corollary 1 §2 10 . □
Pr oposition 1 §3 4 . Let f ∈ H d [ q ] be such that all its coefficients are integers of absolute
value at most H . Then either κ ( f ) = ∞ or
κ ( f ) ≤ √ 2 DNH ( 2 2 − n +1
2 D n +2 H √ N ) 2( n +1)(4 D ) 2( n +1)
= O ( DHN ) O ( D ) 2( n +1)
.

52 Josué Tonelli-Cueto 1 §3
Proof. We take the maximum of all bounds given by Theorem 1 §2 13 noting dim H d [ q ] L ≤
N. □
Pr oposition 1 §3 5 . (A) Let f ∈ H d [ q ] be a KSS random polynomial tuple. Then for t ≥
( n 2 (2 q ) n +1 D n + 1)( N − 1) ,
Ð ( κ ( f ) ≥ t ) ≤ 11( n 2 (2 q ) n +1 D n + 1)( N − 1) 1
t
and
Å log κ ( f ) ≤ log ( N − 1) + n ( log D + 6 log 2) + ( n + 1) log q + log (30) .
(S) Let f ∈ H d [ q ] , σ ∈ [0 , 1] and f σ ∈ H d [ q ] a random polynomial tuple uniformly dis-
tributed in B W ( f , σ ) . Then for t ≥ ( n 2 (2 q ) n +1 D n + 1)( N − 1) σ − 1 ,
Ð ( κ ( f σ ) ≥ t ) ≤ 11( n 2 (2 q ) n +1 D n + 1)( N − 1) σ − 1 1
t
and
Å log κ ( f σ ) ≤ log ( N − 1) + n ( log D + 6 log 2) + ( n + 1) log q + log σ − 1 + log (30) .
Proof. We apply the union bound for a random variable that is the maximum of several
random variables together with Proposition 1 §2 16 . We then observe that dim H d [ q ] L ≤ N
and
# [ q ] ≤ n +1 =
n +1
∑
k =0 ( q
k ) ≤ 2 n q n +1 .
□
Pr oposition 1 §3 6 . (A) Let f ∈ H d [ q ] be a KSS random polynomial tuple and x ∈ Ó n .
Then for t ≥ 2 ,
Ð ( κ ( f , x ) ≥ t ) ≤ 7 n q n +1 ( 119 N
n + 1 ) n +1
2 ( ln 1
2 t
t ) n +1
.
(S) Let f ∈ H d [ q ] , σ > 0 , f σ := f + σ ∥ f ∥ W g be a random polynomial tuple such that
g ∈ H d [ q ] is a KSS random polynomial tuple and x ∈ Ó n . Then for t ≥ 2 ,
Ð ( κ ( f σ , x ) ≥ t ) ≤ 7 n q n +1 ( 119 N
n + 1 ) n +1
2 ( ln 1
2 t
t ) n +1 ( 1 + 1
σ ) n +1
.
Proof. Analogous to the one of Proposition 1 §3 5 , but applying Theorem 1 §2 17 this time. □

1 §4 Condition and Homology in Semialgebraic Geometry 53
1 §4 κ aff : a condition number for affine semialgebraic sets
The traditional way to pass from the affine to the homogenous world is homogeneization.
The homogeneization map
h : P d [ q ] → H d [ q ]
p 7→ p h := ( p i ( X / X 0 ) X d i
0 ) i ∈ [ q ]
,
( 1.28 )
takes each p i to its homogenization adding the variable X 0 . This map allows us to transform
tuples of affine polynomials into tuples of homogeneous polynomials and so transfer the
theory developed for the spherical case to the affine case. In this way, the Weyl norm of
p ∈ P d [ q ] is defined by ∥ p ∥ W := ∥ p h ∥ W and a KSS random polynomial tuple p ∈ P d [ q ] is
defined as a random polynomial tuple such that p h ∈ H d [ q ] is a KSS random polynomial
tuple.
Together with the above map, we consider the diffeomorphism
Ю : Ò n → Ó n
+ := { z ∈ Ó n | z 0 > 0 }
x 7→ Ю ( x ) := 1
√ 1 + ∥ x ∥ 2 ( 1
x ) ,
( 1.29 )
which takes the affine space Ò n onto the upper half of Ó n , Ó n
+ . Note that the maximal circle
Ó n
0 := { z ∈ Ó n | z 0 = 0 } corresponds to the points at infinity of Ò n inside the com-
pactification Ó n
+ ∪ Ó n
0 , which differs from the usual compactification Ð n . The main reason to
compactify in the sphere is that one can still speak of signs of polynomials in Ó n , while this
is not possible, in general, in Ð n .
Given a Boolean formula Φ over p , we can naturally consider the Boolean formula Φ h
over p h obtained by substituting the p i in Φ by their corresponding homogeneization p h
i .
Now, we need to add to the polynomial tuple p h and the formula Φ h a polynomial to encode
the sign of X 0 . To do this, we consider the polynomial tuple
H ( p ) := ( ∥ p ∥ W X 0 , p h ) ( 1.30 )
and the Boolean formula
H (Φ) := Φ h ∧ ( H ( p ) 0 > 0) ( 1.31 )
over it. We can see then that
Ю ( W ( p , Φ)) := S ( H ( p ) , H (Φ)) .
The way we choose the scaling for X 0 in H ( f ) is such that that it has the same weight as p h .
Following the transfer condition, we expect p ∈ P d [ q ] to be well-conditioned if H ( p ) is
so. This motivates the following definition.
Definition 1 §4 1 . Let p ∈ P d [ q ] . The global affine intersection condition number of p ,
κ aff ( p ) , is the quantity in [1 , ∞ ] given by
κ aff ( p ) := κ ( H ( p )) . ( 1.32 )

54 Josué Tonelli-Cueto 1 §4
We note that for this condition number, the following version of Theorem 1 §3 2 holds.
This again can be seen as the justification for calling the above quantity condition number.
Theor em 1 §4 1 . Let p ∈ P d [ q ] be such that κ aff ( p ) < ∞ . Then for all Boolean formula Φ
over f and every g ∈ B W ( p , κ aff ( p ) − 1 min i ∥ p i ∥ W ) , H • ( W ( p , Φ))  H • ( W ( g , Φ)) .
Proof. Since min i ∥ H ( p ) i ∥ W = min i ∥ p i ∥ W and dist W ( H ( p ) , H ( g )) ≤ √ 2 dist W ( p , g ) , it fol-
lows from Theorem 1 §3 2 . □
Before continuing, let H ∞
d [ q ] be the space of d -homogeneous polynomial q -tuples in
the variables X 1 , . . . , X n and consider the orthogonal projection
h : P d [ q ] → H ∞
d [ q ]
p 7→ p h := p h (0 , X 1 , . . . , X n ) ( 1.33 )
that maps each p i to its d i -homogeneous part ( p i ) h , which is the polynomial obtained from
p i by eliminating all terms that are not of degree d i . The behaviour of p at the hyperplane
at infinity Ó n
0 is precisely captured by the behaviour of p h at Ó n − 1 . The following proposition
follows immediately from Proposition 1 §3 1
Pr oposition 1 §4 2 . Let p ∈ P d [ q ] . Then κ aff ( p ) < ∞ iff all the following hold:
(1) For all i ∈ [ q ] , Z ( p i ) ⊆ Ò n and Z Ó (( p i ) h ) ⊆ Ó n − 1 are regular.
(2) For all L ⊆ [ q ] , the intersection ∩ i ∈ L Z ( p i ) is tranversal in Ò n and ∩ i ∈ L Z Ó (( p i ) h )
tranversal in Ó n − 1 . □
Due to the above definition, we can see that the set of ill-posed polynomial tuples
Σ aff
d [ q ] := H − 1 (Σ d [ q ]) = { p ∈ P d [ q ] | κ aff ( p ) = ∞ } decomposes as
Σ aff
d [ q ] = Σ aff
d [ q ] + ∪ Σ aff
d [ q ] 0 ( 1.34 )
with
Σ aff
d [ q ] + := { g ∈ P d [ q ] | g h ∈ Σ d [ q ] } and Σ aff
d [ q ] 0 := { g ∈ P d [ q ] | g h ∈ Σ ∞
d [ q ] }
( 1.35 )
where Σ ∞
d [ q ] := { g ∈ H ∞
d [ q ] | κ ( q ) = ∞ } . Intuitively, Σ aff
d [ q ] + are those polynomial tuples
for which the ill-posedness comes from a non-transversal intersection and Σ aff
d [ q ] 0 those for
which the ill-posedness arrives from a tangency to the hyperplane at infinity. We can get the
following more quantitative statement.
Theor em 1 §4 3 . Let p ∈ P d [ q ] . Then
κ aff ( p ) ≤ max { κ ( p h ) , (2 + 3 D ) max
L ∈ [ q ] ≤ n +1 ∥ p ∥ W κ ( p L
h )
∥ p L
h ∥ W } ≤ (2 + 3 D ) ∥ p ∥ W
dist W ( p , Σ aff
d [ q ])
Lemma 1 §4 4 . Let g ∈ P d [ q ] , α > ∥ g ∥ W and x ∈ Ó n . Then
κ (( α X 0 , g h ) , x ) ≤       
(2 + 3 D ) α κ ( g h , π Ó n
0 ( x ))
∥ g h ∥ W if | x 0 | ≤ √ 2
2
2 if | x 0 | ≥ √ 2
2
.
In particular, κ ( α X 0 , g h ) ≤ (2 + 3 D ) α κ ( g h )
∥ g ∥ W .

1 §4 Condition and Homology in Semialgebraic Geometry 55
Proof of Theorem 1 §4 3 . We observe that, by the weak condition number theorem for κ
(Proposition 1 §3 3 ),
max { κ ( p h ) , (2 + 3 D ) max
L ∈ [ q ] ≤ n +1 ∥ p ∥ W κ ( p L
h )
∥ p L
h ∥ W }
≤ (2 + 3 D ) max { ∥ p ∥ W
dist W ( p h , Σ d [ q ])
, ∥ p ∥ W
dist W ( p h , Σ ∞
d [ q ]) }
= (2 + 3 D ) max { ∥ p ∥ W
dist W ( p , Σ aff
d [ q ] + )
, ∥ p ∥ W
dist W ( p , Σ aff
d [ q ] 0 ) }
= (2 + 3 D ) ∥ p ∥ W
dist W ( p , Σ aff
d [ q ])
.
Hence we only have to prove the first inequality. By the definition of κ aff , we have that
κ aff ( p ) = max
L ∈ [ q ] ≤ n +1 max { κ ( ( p L ) h ) , κ ( ∥ p ∥ W X 0 , ( p L ) h ) } .
Therefore it is enough to show that
κ ( ∥ p ∥ W X 0 , ( p L ) h ) ≤ 4 D ∥ p ∥ W
∥ p h ∥ W
κ ( p h ) .
Now, this is shown by Lemma 1 §4 4 by setting α = ∥ p ∥ W ≥ ∥ p L
h ∥ W and g = p L , and so the
proof concludes. □
Proof of Lemma 1 §4 4 . Let H α ( g ) := ( α X 0 , g h ) , so that ∥ H α ( g ) ∥ W = √ α 2 + ∥ g ∥ 2
W . Since
∥ H α ( g )( x ) ∥ ≥ α | x 0 | , we have
κ ( H α ( g ) , x ) ≤ √ 1 + ∥ g ∥ 2
W
α 2 | x 0 | − 1 ≤ √ 2 | x 0 | − 1 .
This shows the inequality for | x 0 | ≥ 1/ √ 2 . We assume now | x 0 | ≤ 1/2 , so that dist Ó ( x , Ó n
0 ) ≤
π
3 | x 0 | . By the 2nd Lipschitz property (Proposition 1 §2 7 ),
κ ( H α ( g ) , π Ó n
0 ( x )) ≥ κ ( H α ( g ) , x )
1 + π
3 D κ ( H α ( g ) , x ) | x 0 | .
Now, since κ ( H α ( g ) , x ) | x 0 | ≤ √ 2 , this gives
κ ( H α ( g ) , π Ó n
0 ( x )) ≥ κ ( H α ( g ) , x )
1 + √ 2 π
3 D
.
To finish the proof it is enough to show that for y ∈ Ó n
0 ,
κ ( H α ( g ) , y ) ≤ 2 α
∥ g h ∥ W
κ ( g , y ) .
Now, ∥ H α ( g ) ∥ W ≤ √ 2 α , so it is enough to show that
σ q +1 ( α e ∗
0
∆ − 1
d D y g h ) ≥ 1
√ 2 σ q (∆ − 1
d D y g h ) .

56 Josué Tonelli-Cueto 1 §4
Let ( t v ∗ ) with v ∈ Ò q and t 2 + ∥ v ∥ 2 = 1 be such that
σ q +1 ( α e ∗
0
∆ − 1
d D y g h ) =      ( t v ∗ ) ( α e ∗
0
∆ − 1
d D y g h )      .
By an easy computation, we can see that if | t | ≤ 1/ √ 2 ,
     ( t v ∗ ) ( α e ∗
0
∆ − 1
d D y g h )      ≥ 1
√ 2 σ q (∆ − 1
d D y g h ) ,
and if | t | ≥ 1/ √ 2 ,
     ( t v ∗ ) ( α e ∗
0
∆ − 1
d D y g h )      ≥ √ α 2 ( | t | − √ 1 − t 2 ) 2
+ (1 − t 2 ) σ q (∆ − 1
d D y g h ) 2
≥ 1
√ 2 σ q (∆ − 1
d D y g h ) ,
where the inequality follows from direct minimization and the fact that α ≥ σ q (∆ − 1
d D y g h ) ,
which follows from α ≥ ∥ g ∥ W and Corollary 1 §1 7 . □
Remark 1 §4 1 . We observe that the last inequality in Theorem 1 §4 3 is precisely [88 ; Propo-
sition 4.16 ] when D ≥ 2 . However, we note that our proof is different from the one given
in [88] , which, in principle, could be extended to more general conic condition numbers. ¶
Motivated by Theorem 1 §4 3 , let us define
κ ∞
aff ( p , x ) := max
L ∈ [ q ] ≤ n +1 ∥ p ∥ W κ ( p L
h , x )
∥ p L
h ∥ W
( 1.36 )
for p ∈ P d [ q ] and x ∈ Ó n
0 , and κ ∞
aff ( p ) := max y ∈ Ó n
0 κ ∞
aff ( p , y ) . With the above result, we can
perform the usual complexity analyses as shown above for κ . We only sketch the proofs as
they are identical to the ones for κ and κ .
Cor ollary 1 §4 5 . Let p ∈ P d [ q ] be such that all its coefficients are integers of absolute value
at most H . Then either κ aff ( p ) = ∞ or
κ aff ( p ) ≤ √ 2 DNH ( 2 2 − n +1
2 D n +2 H √ N ) 2( n +1)(4 D ) 2( n +1)
= O ( DHN ) O ( D ) 2( n +1)
.
Sketch of proof. By Theorem 1 §4 3 , we just need to bound for κ ( p h ) and κ ∞
aff ( p ) . For the first,
we apply Proposition 1 §3 4 . For the latter, we proceed as in the proofs of Theorem 1 §2 13
and Proposition 1 §3 4 . The bound obtained for κ ∞
aff ( p ) will be like the one above, but with
n − 1 in the place of n . This makes that multiplying by (2 + 3 D ) does not affect the final bound
that we obtain. □
Cor ollary 1 §4 6 . (A) Let p ∈ P d [ q ] be a KSS random polynomial tuple. Then for t ≥
( n 2 (2( q + 1)) n +1 D n + 1)( N − 1) ,
Ð ( κ aff ( p ) ≥ t ) ≤ 55 D ( n 2 (2( q + 1)) n +1 D n + 1)( N − 1) 1
t
and
Å log κ aff ( p ) ≤ log ( N − 1) + ( n + 1)( log D + 9 log 2) + ( n + 1) log ( q + 1) + log (30) .

1 §4 Condition and Homology in Semialgebraic Geometry 57
(S) Let p ∈ P d [ q ] , σ ∈ [0 , 1] and p σ ∈ P d [ q ] a random polynomial tuple uniformly dis-
tributed in B W ( p , σ ) . Then for t ≥ ( n 2 (2( q + 1)) n +1 D n + 1)( N − 1) σ − 1 ,
Ð ( κ aff ( p σ ) ≥ t ) ≤ 55 D ( n 2 (2( q + 1)) n +1 D n + 1)( N − 1) σ − 1 1
t
and
Å log κ aff ( p σ ) ≤ log ( N − 1) + ( n + 1)( log D + log ( q +1 )+9 log 2) + log σ − 1 + log (30) .
Sketch of proof. By Theorem 1 §4 3 , we have to obtain tail bounds for κ ( p h ) and κ ∞
aff ( p ) . For
the first, we apply Proposition 1 §3 5 . For the latter, we proceed like in the proof of Proposi-
tion 1 §3 5 after noting that κ ∞
aff ( p ) ≤ ∥ p ∥ W / dist W (Σ aff
d [ q ] 0 ) . We use a union bound to reduce
from the case of Σ aff
d [ q ] 0 to the case of { p ∈ H d [ q ] | p L ∈ Σ ∞ , L
d [ q ] } . Then we use that
p 7→ p h is an orthogonal projection to apply the degree bound in Proposition 1 §2 15 to the
latter sets. Finally, we use Theorem 1 §2 14 . □
The following probabilistic bound will be useful later.
Cor ollary 1 §4 7 . (A) Let p ∈ P d [ q ] be a KSS random polynomial tuple and x ∈ Ó n
0 . Then
for t ≥ 2 ,
Ð ( κ ∞
aff ( p , x ) ≥ t ) ≤ 7 n q n +1 ( 119 N
n ) n
2 ( ln 1
2 t
t ) n
.
(S) Let p ∈ P d [ q ] , σ > 0 , p σ := p + σ ∥ p ∥ W g be a random polynomial tuple such that
g ∈ H d [ q ] is a KSS random polynomial tuple and x ∈ Ó n
0 . Then for t ≥ 2 ,
Ð ( κ ∞
aff ( p σ , x ) ≥ t ) ≤ 7 n q n ( 119 N
n ) n
2 ( ln 1
2 t
t ) n ( 1 + 1
σ ) n
.
Sketch of proof. To handle
∥ p ∥ W
√ ∥ p L
h ( x ) ∥ 2 + σ q (∆ − 1
d D x p L
h ) 2
,
we separate numerator and numerator as in the proof of Theorem 1 §2 17 . Then the rest is the
same, except that we have just n variables now, instead of n + 1 . To handle the maximum,
we just apply the union bound as in the proof of Proposition 1 §3 6 . □
Further comments
Many of the results in this chapter can be found in [87] and [88, 91] . However, there are
some important additions: the gap theorem (Theorem 1 §2 13 , Proposition 1 §3 4 and Corol-
lary 1 §4 5 ) for integer polynomial tuples, the tail bounds coming from geometric functional
analysis (Theorems 1 §2 17 and 1 §2 19 , Proposition 1 §3 6 and Corollary 1 §4 7 ), and the in-
equality of Theorem 1 §4 3 .
In addition to the new results, our presentation differs from those in [87] and [88, 91] .
On the one hand, our definition of κ is with the q th singular value, instead of the µ -condition

58 Josué Tonelli-Cueto 1 §4
or the operator norm. This change leads to a clearer form of κ which is easier to parse.
We introduce µ afterwards, but only after familiarity with κ has been attained. On the other
hand, our focus goes away from the condition number theorem, and more into the properties
and possibles bounds of κ . Even though, this means that we acknowledge the beauty of
a condition number theorem and the geometric interpretation it gives κ . We don’t view this
as the center of the theory, since from an algorithmic point of view the other properties are
more important than a ‘fancy’ geometric interpretation. This should be viewed as a break
with the philosophy of [87] .
It remains an important exercise to develop the above theory in the multihomogeneous
setting, meaning that κ ( f , x ) and κ ( f ) should be invariant under the scaling of each poly-
nomial f i in f . This development would lead to a more robust κ . The reason for this is that
semialgebraic sets are defined with atoms involving only one atom at a time.
We note that our approach to condition numbers follows the philosophy of the worst
variation. It would be interesting to study weak variations, where we consider the high-
probability-variation, following the notion of weak condition number introduced by Lotz and
Noferini [282] .

59
There are no dogmas to which we must conform. Our program is simple: to give numerical
meaning to as much as possible of classical abstract analysis.
Errett Bishop, Foundations of Constructive Analysis
2
Dif fer ential semialgebraic geometry
with condition-based inequalities
In semialgebraic geometry, it is usual to have results depending on “weak” inequalities.
These inequalities are of the form “for sufficiently small a ” or “for x sufficiently smaller than
y ”. Unfortunately, from an applied and computational viewpoint, these statements can be
useless, because they don’t give explicit bounds that can be used to obtain numbers sat-
isfying the desired statements. In the symbolic world, one can solve the issue by adding
infinitesimals; in the numerical world, we don’t have the luxury of using infinitesimals. Thus
we need to make the “weak” inequalities explicit and find explicit values for them.
In this chapter, we will substitute such weak inequalities by strong inequalities depending
on the condition number in the case of two theorems: Durfee’s theorem (Theorem 2 §3 2 ) and
Gabrielov-Vorobjov approximation theorem (Theorem 2 §4 2 ). Or, paraphrasing Bishop [Q4] ,
we will give “numerical meaning” to these inequalities in the well-posed case. The first result
will be fundamental for our constructions of simplicial complexes, and the second one for
passing from the arbitrary case to the closed case.
First, we recall Newton’s vector field and use a discontinuous version of it to prove a
converse of the Exclusion Lemma (Corollary 1 §1 8 ); second, we present the Thom-Mather
theory that will play a fundamental role in this thesis; third, we introduce our main technical
tool, ( f , λ ) -lartitions and ( f , λ ) -partitions; and four and last, we prove, respectively, Durfee’s
and Gabrielov-Vorobjov approximation theorem.
2 §1 A converse to the Exclusion Lemma
Given any smooth map f : Ó n → Ò q , we consider the open set Ω f := { x ∈
Ó n | D x f is surjective } and, on it, the Newton vector field of f as the vector field given by
N f
x := − D x f † f ( x ) . ( 2.1 )

60 Josué Tonelli-Cueto 2 §1
The main property of this vector field is that for any integral path t 7→ z t ,
f ( z t ) = f ( z 0 ) e − t . ( 2.2 )
This property follows from the chain rule and the properties of the pseudoinverse.
Recall that B Ó and B Ó denote, respectively, the open and closed balls in Ó n with respect
the geodesic distance and that, for r > 0 , the spherical r - neighborhood of X ⊂ Ó n is the
set
U Ó ( X , r ) := { p ∈ Ó n | d Ó ( p , X ) ≤ r } = ∪
x ∈ X
B Ó ( x , r ) . ( 2.3 )
For f ∈ H d [ q ] , the algebraic neighborhood of Z Ó ( f ) with tolerance r is the set
Z Ó
r ( f ) := { x ∈ Ó n | ∥ f ( x ) ∥ ≤ r ∥ f ∥ W } . ( 2.4 )
The following theorem is a two-way version of the Exclusion Lemma (Corollary 1 §1 8 ) for
algebraic sets.
Pr oposition 2 §1 1 . Let f ∈ H d [ q ] and r > 0 be such that √ 2 κ ( f ) r < 1 . Then
(a) Z Ó ( f ) ⊆ Z Ó
r ( f ) ,
(b) U Ó ( Z Ó ( f ) , r ) ⊆ Z Ó
D 1/2 r ( f ) , and
(c) Z Ó
r ( f ) ⊆ U Ó ( Z Ó ( f ) , √ 2 κ ( f ) r ) .
Proof. (a) is obvious and (b) is just a reformulation of the Exclusion Lemma (Corollary 1 §1 8 ).
(c). Take x ∈ Z Ó
r ( f ) and consider the integral path t 7→ x t of the Newton vector field
of f starting at x . Since √ 2 κ ( f ) r < 1 , we have that Z Ó
r ( f ) ⊆ Ω f , by Proposition 1 §2 3 , and
so the Newton vector field is defined at every point of Z Ó
r ( f ) . By ( 2.2 ), t 7→ x t does not
leave Z Ó
r ( f ) and so it can be extended indefinitely obtaining a global integral path [0 , ∞ ) ∋
t 7→ x t .
We have that
∥ ˙
x t ∥ ≤ ∥ D x t f † ∥ ∥ f ( x t ) ∥ = ∥ D x t f † ∥ ∥ f ( x ) ∥ e − t = ∥ f ∥ W
σ q ( D x t f ) ∥ f ( x ) ∥
∥ f ∥ W
e − t ≤ √ 2 κ ( f ) r e − t ,
where the first inequality follows from ˙
x t = − D x t f † f ( x t ) , which follows from ( 2.1 ); the
first equality from ( 2.2 ), the second equality from the form of the singular values of the
pseudoinverse, and the second inequality from Proposition 1 §2 3 . Therefore
∫ ∞
0 ∥ ˙
x t ∥ d t < √ 2 κ ( f ) r
and so x t converges absolutely and lim t →∞ x t exists. By ( 2.2 ), this limit belongs to Z Ó ( f ) .
We have thus shown that starting from x we can reach a point of Z Ó ( f ) following a path in Ó n
of length less than √ 2 κ ( f ) r . Hence dist Ó ( x , Z Ó ( f )) ≤ √ 2 κ ( f ) r and the claim follows. □
In the semialgebraic case, we can prove an analog of Proposition 2 §1 1 using a discon-
tinuous generalization of the Newton vector field.

2 §1 Condition and Homology in Semialgebraic Geometry 61
2 §1 -1 Boolean formulas over ( f , t ) and algebraic neighborhoods
We introduce several geometric notions that will be central later on. First, we extend our
universe of considered functions from homogeneous polynomials to homogeneous poly-
nomial with constants added. The reason for this is that these polynomials appear in the
Gabrielov-Vorobjov construction (see 2 §4 ) and so the theory has to be extended to include
them.
Definition 2 §1 1 . Let f ∈ H d [ q ] and t ∈ Ò e , a Boolean formula over ( f , t ) is a Boolean
formula Φ supported on
{ ( f i = ∥ f i ∥ W t j ) , ( f i , ∥ f i ∥ W t j ) ,
( f i > ∥ f i ∥ W t j ) , ( f i ≥ ∥ f i ∥ W t j ) , ( f i < ∥ f i ∥ W t j ) , ( f i ≤ ∥ f i ∥ W t j ) | i ∈ [ q ] , j ∈ [ e ] } .
Given a Boolean formula Φ over ( f , t ) , the realization of ( f , t , Φ) , S ( f , t , Φ) , is the semial-
gebraic set
S ( f , t , Φ) := Φ Ó n ( ˆ
f − 1
i ( t j ) , ˆ
f − 1
i ( Ò \ t j ) ,
ˆ
f − 1
i ( t j , ∞ ) , ˆ
f − 1
i [ t j , ∞ ) , ˆ
f − 1
i ( −∞ , t j ) , ˆ
f − 1
i ( −∞ , t j ] | i ∈ [ q ] , j ∈ [ e ] ) ( 2.5 )
where ˆ
f = ( f i / ∥ f i ∥ W ) i ∈ [ q ] and the subscript in Φ Ó n indicates that we evaluate subsets of
the sphere. In other words, S ( f , t , Φ) is the spherical semialgebraic set obtained interpreting
( f , t , Φ) in the obvious way.
Remark 2 §1 1 . In the above definition, t ∈ Ò e gives us the constants that we can modify our
original polynomial tuple with. ¶
Now, we introduce some notions for formulas and we define algebraic neighborhoods
for closed semialgebraic sets.
Definition 2 §1 2 . Let f ∈ H d [ q ] and t ∈ Ò e . Then:
(m) a monotone formula over ( f , t ) is a Boolean formula over ( f , t ) that contains no nega-
tions.
(l) a lax formula over ( f , t ) is a monotone formula over ( f , t ) whose atoms are of the
form ( f i = ∥ f i ∥ W t j ) , ( f i ≥ ∥ f i ∥ W t j ) and ( f i ≤ ∥ f i ∥ W t j ) .
(pc) a purely conjunctive formula over ( f , t ) is a monotone formula over ( f , t ) that does
not contain disjunctions, i.e., it is a formula of the form ∧ i ∈ I ( f a i ∝ i ∥ f a i ∥ W t b i ) where
a ∈ [ q ] I , b ∈ [ e ] I and ∝ ∈ { = , , , >, ≥ , <, ≤ } I .
Definition 2 §1 3 . Let f ∈ H d [ q ] , t ∈ Ò e , r > 0 and ϕ be a lax formula over ( f , t ) . The alge-
braic neighborhood of S ( f , t , ϕ ) with tolerance r , S r ( f , t , ϕ ) , is the spherical semialgebraic
set given by
S r ( f , t , Φ) := Φ Ó n ( ˆ
f − 1
i [ t j − r , t j + r ] , ˆ
f − 1
i [ t j − r , ∞ ) , ˆ
f − 1
i ( −∞ , t j + r ] | i ∈ [ q ] , j ∈ [ e ] ) .
( 2.6 )

62 Josué Tonelli-Cueto 2 §1
In other words, S r ( f , t , Φ) is the spherical semialgebraic set obtained by substituting in Φ the
atoms ( f i = ∥ f i ∥ W t j ) by ( f i ≤ ∥ f i ∥ W ( t j + r )) ∧ ( f i ≥ ∥ f i ∥ W ( t j − r )) , the atoms ( f i ≥ ∥ f i ∥ W t j )
by ( f i ≥ ∥ f i ∥ W ( t j − r )) and the atoms ( f i ≤ ∥ f i ∥ W t j ) by ( f i ≤ ∥ f i ∥ W ( t j + r )) and interpreting
the obtained formula in the obvious way.
To control the well-posedness of ( f , t ) , we have to consider the separation , Щ ( t ) , of
t ∈ Ò e given by
Щ ( t ) := inf
i , j | t i − t j | ( 2.7 )
in addition to the condition number κ . This parameter allows us to control that certain inter-
sections, such as S ( f , t , ( f i ≥ ∥ f i ∥ W t j ) ∧ ( f i ≤ ∥ f i ∥ W t k )) , with t j > t k , remains empty when
passing to algebraic neighborhoods with sufficiently small tolerance. The following technical
proposition makes this clear.
Pr oposition 2 §1 2 . Let f ∈ H d [ q ] , t ∈ Ò e and r > 0 be such such that Щ ( t ) > 2 r . Then:
1. For every purely conjunctive lax formula ϕ over ( f , t ) , there exist a purely conjunctive
lax formula NF ( ϕ ) , called normal form of ϕ , of the form
NF ( ϕ ) ≡ ∧
и ∈ И +
( f и ≥ t α ( и ) ∥ f и ∥ W ) ∧ ∧
и ∈ И −
( f и ≤ t α ( и ) ∥ f и ∥ W )
∧ ∧
и ∈ И 0
(( f и ≥ t l b ( и ) ∥ f и ∥ W ) ∧ ( f и ≤ t u b ( и ) ∥ f и ∥ W ))
with И + , И − , И 0 ⊆ [ q ] pairwise disjoint, α : И + ∪ И − → [ e ] and l b , u b : И 0 → [ e ]
such that for all и ∈ И 0 , t l b ( и ) ≤ t u b ( и ) , and such that
S ( f , t , ϕ ) = S ( f , t , NF ( ϕ )) and S r ( f , t , ϕ ) = S r ( f , t , NF ( ϕ )) .
2. For every lax formula Φ over ( f , t ) , there exist a lax formula DNF (Φ) , called disjunctive
normal form of Φ , of the form
DNF (Φ) ≡ ∨
ξ ∈ Ξ
ϕ ξ
with ϕ ξ purely conjunctive and lax in normal form, and such that
S ( f , t , Φ) = S ( f , t , DNF (Φ)) and S r ( f , t , Φ) = S r ( f , t , DNF (Φ)) .
Proof. 1. Since ∧ is commutative, in the sense that permuting atoms do not affect the real-
ization, we can just focus in the case
∧
k ∈ K
( f i ∝ k t a ( k ) ∥ f i ∥ W )
with a : K → [ e ] and ∝ ∈ { ≥ , ≤ , = } K , which is obtained when gathering all atoms in which
a particular f i appears. Further, by splitting atoms of the form ( f i = t a ( k ) ∥ f i ∥ W ) into ( f i ≤
t a ( k ) ∥ f i ∥ W ) ∧ ( f i ≥ t a ( k ) ∥ f i ∥ W ) , which again does not change the realization at all, we can
assume that ∝ ∈ { ≤ , ≥ } K .

2 §1 Condition and Homology in Semialgebraic Geometry 63
Now, we observe that substituting, respectively,
∧
k ∈ K
∝ k is ≤
( f i ≤ t a ( k ) ∥ f i ∥ W ) by ©  « f i ≤ ©  « min
k ∈ K
∝ k is ≤
t a ( k ) ª ® ¬ ∥ f i ∥ W ª ® ¬
and ∧
k ∈ K
∝ k is ≥
( f i ≥ t a ( k ) ∥ f i ∥ W ) by ©  « f i ≥ ©  « max
k ∈ K
∝ k is ≥
t a ( k ) ª ® ¬ ∥ f i ∥ W ª ® ¬
does not change the realization. Hence we have substituted our initial factor by a factor of
the form ( f i ≤ t α ( i ) ∥ f i ∥ W ) , ( f i ≥ t α ( i ) ∥ f i ∥ W ) or (( f i ≥ t l b ( i ) ∥ f i ∥ W ) ∧ ( f i ≤ t u b ( i ) ∥ f i ∥ W )) . In
the first two cases, there is nothing to prove. In the last case, if t l b ( j ) > t u b ( j ) , then both
S ( f , t , Φ) and S r ( f , t , Φ) are empty, since Щ ( t ) > 2 r , and we can just take NF ( ϕ ) to be
the empty formula. The claim is proven.
2. Using that ∧ is distributive with respect ∨ , in the sense that passing from Φ 0 ∧ (Φ 1 ∨ Φ 2 )
to (Φ 0 ∧ Φ 1 ) ∨ (Φ 0 ∧ Φ 2 ) does not affect the realization, we can transform Φ into a formula
of the form ∨
ξ ∈ Ξ
ϕ ξ
with ϕ ξ purely conjunctive and such that
S ( f , t , Φ) = S ©  « f , t , ∨
ξ ∈ Ξ
ϕ ξ ª ® ¬ and S r ( f , t , Φ) = S r ©  « f , t , ∨
ξ ∈ Ξ
ϕ ξ ª ® ¬ .
Applying (1) to each ϕ ξ finishes the proof. □
Remark 2 §1 2 . We observe that Proposition 2 §1 2 does not necessarily give efficient algo-
rithms. For example, DNF (Φ) can have size exponential in the size of Φ as one can sees by
slightly modifying Example 0 §1 3 . ¶
2 §1 -2 Discontinuous Newton vector field
We can now state and prove the semialgebraic version of Proposition 2 §1 1 .
Pr oposition 2 §1 3 . Let f ∈ H d [ q ] , t ∈ ( − T , T ) e and r > 0 be such that √ 2 κ ( f )( r + T ) < 1
and Щ ( t ) > 2 r . Then, for every lax formula Φ over ( f , t ) ,
(a) S ( f , t , Φ) ⊆ S r ( f , t , Φ) ,
(b) U Ó ( S ( f , t , Φ) , r ) ⊆ S D 1/2 r ( f , t , Φ) , and
(c) S r ( f , t , Φ) ⊆ U Ó ( S ( f , t , Φ) , √ 2 κ ( f ) r ) .
We observe that (a) is trivial and that (b) follows immediately from the Exclusion Lemma
(Corollary 1 §1 8 ). Therefore we will focus on (c). By Proposition 2 §1 2 , we see that it is enough

64 Josué Tonelli-Cueto 2 §1
to prove the above proposition for a purely conjunctive lax formula ϕ in normal form, which
has the form
ϕ ≡ ∧
и ∈ И +
( f и ≥ t α ( и ) ∥ f и ∥ W ) ∧ ∧
и ∈ И −
( f и ≤ t α ( и ) ∥ f и ∥ W )
∧ ∧
и ∈ И 0
(( f и ≥ t l b ( и ) ∥ f и ∥ W ) ∧ ( f и ≤ t u b ( и ) ∥ f и ∥ W ))
with И + , И − , И 0 ⊆ [ q ] pairwise disjoint, α : И + ∪ И − → [ e ] and l b , u b : И 0 → [ e ] such that
for all и ∈ И 0 , t l b ( и ) ≤ t u b ( и ) . We know define the following set-valued maps
Ó n ∋ x 7→ L + ( x ) := { и ∈ И + | f и ( x ) ≤ t α ( и ) ∥ f и ∥ W } ,
Ó n ∋ x 7→ L − ( x ) := { и ∈ И − | f и ( x ) ≥ t α ( и ) ∥ f и ∥ W } ,
Ó n ∋ x 7→ L l b ( x ) := { и ∈ И 0 | f и ( x ) ≤ t l b ( и ) ∥ f и ∥ W } , and
Ó n ∋ x 7→ L u b ( x ) := { и ∈ И 0 | f и ( x ) ≥ t u b ( и ) ∥ f и ∥ W } .
For each x ∈ Ó n , L + ( x ) , L − ( x ) , L l b ( x ) and L u b ( x ) are pairwise disjoint and each of these
sets encodes the clauses of ϕ that x does not satisfy or could stop satisfying after a small
perturbation.
Consider also the maps Ó n ∋ x 7→ L ( x ) := L − ( x ) ∪ L + ( x ) ∪ L l b ( x ) ∪ L u b ( x ) and
Ó n ∋ x 7→ т ( x ) ∈ Ò L ( x ) given by
т l ( x ) :=           
t α ( l ) ∥ f l ∥ W , if l ∈ L + ( x ) ∪ L − ( x )
t l b ( l ) ∥ f l ∥ W , if l ∈ L l b ( x )
t u b ( l ) ∥ f l ∥ W , if l ∈ L u b ( x )
.
With the help of these maps, we define the discontinuous Newton vector field of ( f , t , ϕ )
N f , t , ϕ
x := − ( D x f L ( x ) ) † ( f L ( x ) ( x ) − т ( x )) ( 2.8 )
where D x f L ( x ) = ( D x f i ) i ∈ L ( x ) selects the rows in D x f L ( x ) indexed by L ( x ) . In general, this
vector field is not continuous, but its solutions are well-behaved. With this vector field, we
can prove Proposition 2 §1 3 .
Lemma 2 §1 4 . (i) Given x 0 ∈ S r ( f , t , ϕ ) , the integral path [0 , T ) ∋ t 7→ x t of the Newton
vector field of f L ( x 0 ) − т ( x 0 ) , N ( f L ( x 0 ) − т ( x 0 )) , starting at x 0 , agrees locally with the integral
path of the discontinuous Newton vector field of ( f , t , ϕ ) , N f , t , ϕ , starting at x 0 .
(ii) Let [0 , T ) ∋ t 7→ x t be an integral path of N f , t , ϕ and t , t ′ ∈ [0 , T ) . If t ′ > t , then
L + ( x t ′ ) ⊇ L + ( x t ) , L − ( x t ′ ) ⊇ L − ( x t ) , L l b ( x t ′ ) ⊇ L l b ( x t ) and L u b ( x t ′ ) ⊇ L u b ( x t ) .
(iii) Given x 0 ∈ S r ( f , t , ϕ ) , there is a forward-time integral path of the discontinuous Newton
vector field N f , t , ϕ of ( f , t , ϕ ) starting at x 0 that extends indefinitely, i.e., for any time.
Remark 2 §1 3 . Although we could prove uniqueness of time-forward integral paths in Lemma
2 §1 4 , we don’t provide such a proof. The main reason for this is that such a uniqueness result
is not needed. ¶

2 §1 Condition and Homology in Semialgebraic Geometry 65
Proof of Proposition 2 §1 3 (c). We restrict to the case of a formula ϕ as described above,
since it is enough to consider this case.
Let x ∈ S r ( f , t , ϕ ) . By Lemma 2 §1 4 , we can consider the integral path of t 7→ x t
starting at this point that extends indefinitely. Further, for all t > 0 , we have that
f l ( x t ) = { т l ( x 0 )+( f i ( x 0 ) − т l ( x 0 )) e − t if l ∈ L ( x 0 )
т l ( x t ) if l ∈ L ( x t ) \ L ( x 0 ) ( 2.9 )
since, if l ∈ L ( x 0 ) , this follows from the formula for N f , t , ϕ , and, if not, then when we add l ,
this holds because f l ( x t ) = т l ( x ) , and so no variation occurs. Because of ( 2.9 ), the integral
path remains in S r ( f , t , ϕ ) . Arguing as in the proof of Proposition 2 §1 1 , we can see that, by
the regularity inequality (Proposition 1 §3 3 ),
∥ ˙
x t ∥ ≤ √ 2 κ ( f ) r e − t .
Therefore ∫ ∞
0 ∥ ˙
x t ∥ d t ≤ √ 2 κ ( f ) r
and so the limit lim t →∞ x t exists and belongs to S ( f , Φ) , by ( 2.9 ). Now, such a path lies on
Ó n and has length at most √ 2 κ ( f ) r . Hence the claim follows. □
Proof of Lemma 2 §1 4 . (i). For the considered integral path, we can check that for t suffi-
ciently small, L ( x t ) = L ( x 0 ) . Indeed, for l ∈ L ( x 0 ) ,
f l ( x t ) = т l ( x 0 )+( f l ( x 0 ) − т l ( x 0 )) e − t
and so l ∈ L ( x t ) for t > 0 , since the inequalities defining L ( x 0 ) will still hold. And for
l ∈ [ q ] \ L ( x 0 ) , we have strict inequalities, and so, by continuity, l ∈ [ q ] \ L ( x t ) for t
sufficiently small. Hence, along the integral path t 7→ x t , N f , t , ϕ
x t = N f L ( x 0 ) − т ( x 0 )
x t for sufficiently
small t . Thus the claim follows.
(ii). We prove the claim only for L + , since for the rest the proof is analogous. By (i) and
the formula for f l ( x t ) above, we can see that f l ( x s ′ ) − т l ( x s ′ ) = ( f l ( x s ) − т l ( x s )) e s ′ − s for
any s , s ′ ∈ [ t , t ′ ) with s ′ > s sufficiently near to s and l ∈ L ( x s ) . This means that
{ s ∈ [ t , t ′ ] | for all s ′ ∈ [0 , s ] , L + ( x s ′ ) ⊇ L + ( x t ) }
is open. Since the defining conditions of L + ( x ) are closed, it is also closed. Thus it agrees
with [ t , t ′ ] and the claim follows for L + .
(iii). By the regularity inequality (Proposition 1 §3 3 ) and (i), we can guarantee that a local
time-forward integral path starting at x 0 exists and that it does not leave S r ( f , t , ϕ ) . By (ii),
we only have to paste finitely many integral paths of the Newton vector field of f L ( x t 0 ) − т ( x t 0 ) .
Hence we can extend the integral path indefinitely, as desired. □
One may think that the proof above can be adapted to obtain a continuous retraction
of S r ( f , t , ϕ ) onto S ( f , t , ϕ ) when ϕ is a purely conjunctive lax formula, so that one proves
Durfee’s theorem (Theorem 2 §3 2 ). However, as shown in Example 2 §1 1 below, the flow
of the discontinuous Newton vector field is not continuous in general. This phenomenon
motivates the introduction of Mather-Thom theory to be able to work with vector fields better
suited for the semialgebraic setting.

66 Josué Tonelli-Cueto 2 §2
Example 2 §1 1 . We consider a pointed cone C (for simplicity, in Ò 2 ) given as ℓ 1 ≥ 0 ∧ ℓ 2 ≥ 0
where ℓ 1 , ℓ 2 are linear functions. In this case, the Newton vector field (over either ℓ 1 = 0 ,
ℓ 2 = 0 , or ℓ 1 = ℓ 2 = 0 ) is just the orthogonal projection and the discontinuous Newton field
is the orthogonal projection onto the correspondent pieces of the boundary. Figure 2 §1 1
shows two such situations for cones with different openings. For S ⊂ { 1 , 2 } the region R S
in the figure is the set { x | L ( x ) = S } .
γ 3
γ 1
γ 2
R { 1 }
R { 1 , 2 }
R { 2 }
ℓ 1 ℓ 2
C
γ 3
γ 2
γ 1
R { 1 }
R { 1 , 2 }
R { 2 }
ℓ 1 ℓ 2
C
Figure 2 §1 1 : Discontinuous Newton vector field for a convex cone
We observe that in the left-hand drawing the flow of the discontinuous Newton vector
field is continuous, while in the right-hand drawing it is not (as illustrated by the integral paths
γ 1 and γ 2 whose end points are far away even though their initial points are near). We also
observe that this difference is not caused by conditioning as κ ( ℓ 1 , ℓ 2 ) is the same for both
situations (each pair of lines being obtained from the other by a rotation). △
2 §2 Mather-Thom theory and some Whitney stratifications
Let us start with a motivation. Gradient retractions are central in Morse theory, where
they are used to establish homotopy equivalences between fibers of Morse functions at pairs
of regular values without critical values in between.
More precisely, it is known that for a submersion α : M → I from a compact manifold
M to an interval I ⊆ Ò , the gradient of α induces a homotopy equivalence α − 1 ( t ) ⊆ α − 1 ( J )
for any subinterval J ⊆ I and any t ∈ J. In more general terms, but also using the gradient of
α to prove it, this translates into the following statement (a particular case of Ehresmann’s
Lemma): for a submersion α : M → I from a compact manifold M to an interval I ⊆ Ò ,
the map α : M → I is a trivial fiber bundle. Recall that a trivial fiber bundle α : E → B is a
continuous map of topological spaces for which there is a subspace F of E (the fiber ) and a
homeomorphism h : E → F × B such that the diagram
E F × B
B
h
α π B
commutes. That is, α is a projection in disguise.

2 §2 Condition and Homology in Semialgebraic Geometry 67
The extension of these results to a more general class of maps is part of the so-called
Mather-Thom theory [190, 285, 388] 1 , which allows one to generalize the results above
from smooth to semialgebraic, not necessarily smooth, maps. Below, we outline the main
notions of this theory, stating a version of the so-called Thom’s first isotopy lemma; show
the theory in action by proving Theorems 1 §2 2 and 1 §3 2 , as promised in the last chapter;
and introduce the main technical construction of this chapter.
2 §2 -1 Whitney stratifications and Thom’s first isotopy lemma
The following definition generalizes the notion of a triangulation of M , by allowing to
decompose M into more general pieces.
Definition 2 §2 1 . [190 ; Ch. I, §1 ] A Whitney stratification of a subset Ω of a smooth manifold
M of dimension m is a partition W of Ω into locally closed smooth submanifolds of M ,
called strata , such that:
F (Locally finite) Every x ∈ Ω has a neighborhood intersecting finitely many strata only.
W (Whitney’s condition b) For every strata ς , σ ∈ W , every point x ∈ ς ∩ σ , every sequence
of points { x ℓ } ℓ ∈ Î in ς converging to x , and every sequence of points { y ℓ } ℓ ∈ Î in σ
converging to x , we have that, in all local charts of M around x ,
lim
ℓ →∞ x ℓ , y ℓ ⊆ lim
ℓ →∞ T y ℓ σ ,
provided both limits exist. The inclusion should be interpreted in the local coordinates
of the chart: x ℓ , y ℓ denotes the straight line joining x ℓ and y ℓ , T y ℓ σ denotes the affine
plane tangent to σ at y ℓ , and the limits are to be interpreted in the corresponding
Grassmannians of Ò m .
A Whitney stratified set (Ω , W ) of M is subset Ω of M together with a Whitney stratifica-
tion W .
Remark 2 §2 1 . In many references, e.g., [285 ; §5 ] , it is usual for the definition of Whitney
stratification to include the so-called boundary condition which states that for every pair of
strata ς , σ ∈ W , ς ∩ σ , ∅ implies ς ⊆ σ . We omit it from the given definition, because this
condition is not needed and “is something of an embarrassment, since it is not preserved
under natural operations on stratifications” [190 ; pp. 16-17 ] . ¶
Remark 2 §2 2 . We note that Whitney’s condition b has not to be checked for every local
chart of M , since it holds for all local charts if it holds for just one local chart of M [285 ;
Lemma 2.2 ] . Further, one can check it in the local chart of an ambient manifold containing
M .
1 The core of the theory was introduced by Thom in 1969 [388] . However, the original paper was hard to
read. In the Spring of 1970, Mather gave a course at Harvard. The lecture notes of this course became the
unofficial reference to the theory, since they explained and expanded in great detail the original ideas of Thom.
In 2012, these references were printed officially [285] . Although we used the more accessible book [190] , we
call the theory ‘Mather-Thom theory’ to acknowledge the two creators of the theory. However, in some other
references like the original reference by Mather [285] , this theory is called Thom-Whitney theory.

68 Josué Tonelli-Cueto 2 §2
We also note that Whitney’s condition b implies the weaker Whiney’s condition a [285 ;
Definition 1.1 ] which states that for every ς , σ ∈ W , x ∈ ς ∩ σ and sequence { y ℓ } in
σ converging to x , we have T x ς ⊆ lim ℓ →∞ T y ℓ σ whenever the limit exists [285 ; Proposi-
tion 2.4 ] . ¶
We go through some examples and non-examples to get familiar with the introduced
concepts.
Example 2 §2 1 . For every smooth manifold M , ( M , { M } ) , is a Whitney stratified set of M .
We will refer to this as the trivial stratified set of M . △
Example 2 §2 2 . The sign map sgn : Ò m → { − 1 , 0 , +1 } m which maps each x ∈ Ò m to the
vector of its signs induces a Whitney stratification on Ò m , which it’s called the sign partition .
We note that the Whitney partitions that we will be working with look locally like the sign
partition. △
Example 2 §2 3 (Spiral) . Consider the stratification of Ò 2 consisting of the point { 0 } , the
smooth one-dimensional submanifold C := { ( e t cos ( t ) , e t sin ( t )) | t ∈ Ò } , and the open
subset σ := Ò 2 \ ( { 0 } ∪ C ) . This stratification does not satisfy Whitney’s condition b.
Note that C is a logarithmic spiral and that the angle between 0 , x and T x C is π /4 for
all x ∈ C. This implies that lim ℓ →∞ 0 , y ℓ ⊈ lim ℓ →∞ T y ℓ C for all sequences { y ℓ } of points
in C, whenever the two limits of lines exist. Therefore Whitney’s condition b cannot hold at
0 ∈ { 0 } ∩ C.
The intuitive reason for this violation of Whitney’s condition b is that the spiral C oscillates
too much around 0 . This means that we should see Whitney’s condition b as a “smoothness
condition” for stratifications, which guarantees that the different strata “paste” nicely to each
other. △
Example 2 §2 4 (Whitney’s umbrella) . [190 ; Ch. I, §1 ] . Consider the algebraic set Ω :=
{ ( x , y , z ) ∈ Ò 3 | x 2 − z y 2 = 0 } , which is known as Whitney’s umbrella . An initial stratifi-
cation of Ω can be obtained separating the line L = { ( x , y , z ) ∈ Ò 3 | x = y = 0 } from
the surface S := Ω \ L. However, one can check that the stratification { L , S } of Ω does not
satisfy Whitney’s condition b at the origin.
Intuitively, the reason for this is different from that for the spiral. The Whitney umbrella
does not have wild variations at the origin. However, one can check that Ω looks different
locally around (0 , 0 , t ) ∈ L depending on whether t < 0 , t = 0 and t > 0 . If t < 0 , Ω
looks locally like a line; if t > 0 , like two planes intersecting transversely; and if t = 0 , like an
umbrella broken by the wind. This allows to see the failure of Whitney’s condition b as the
existence of a radical change in the local topology of Ω as we move along L, which again
can be seen as a lack of “smoothness” of the stratification { L , S } .
However, the stratification { L , S } can be turned into a Whitney stratification by dividing
the line L into O = { (0 , 0 , 0) } , L + := { (0 , 0 , z ) ∈ L | z > 0 } and L − := { (0 , 0 , z ) ∈ L |
z < 0 } . Indeed, { O , L + , L − , S } is a Whitney stratification of Ω and the phenomenon above
does not happen. This procedure can be done in general for semialgebraic sets and one
can show that every semialgebraic set admits a Whitney stratification [190 ; Ch. I, (2.7) ] . △
The following proposition shows that Whitney stratified sets (and Whitney stratifications)
are closed under many of the usual operations.

2 §2 Condition and Homology in Semialgebraic Geometry 69
Pr oposition 2 §2 1 . [190 ; Ch. I, (1.2), (1.3) and (1.4) ] . Let I be a finite set.
(R) Let (Ω , W ) be a locally closed Whitney stratified set of a smooth manifold M . If U is
an open subset of M , then (Ω ∩ U , W | U ) , where W | U := { σ ∩ U | σ ∩ U , ∅ } , is a
Whitney stratified set of M .
(P) For each i ∈ I , let (Ω i , W i ) be a locally closed Whitney stratified set of a smooth
manifold M i . Then ( ∏ i ∈ I Ω i , ∏ i ∈ I W i ) where ∏ i ∈ I W i := { ∏ i ∈ I σ i | σ i ∈ W i } is a
Whitney stratified set of ∏ i ∈ I M i .
(I) Let M be a smooth manifold and, for each i ∈ I , let (Ω i , W i ) be a locally closed Whit-
ney stratified set of M . If W 1 , . . . , W r are transversal, i.e., for every σ 1 ∈ W 1 , . . . , σ r ∈
W r , ∩ r
i =1 σ i is a transversal intersection, then ( ∩ i ∈ I Ω i , ∧ i ∈ I W i ) where
∧
i ∈ I W i := { ∩ i ∈ I σ i | σ i ∈ W i }
is a Whitney stratified set M . □
Recall that a map is proper when its inverse image of any compact subset is compact.
Let A be a partition of A and B one of B, by a stratified homeomorphism f : ( A , A ) →
( B , B ) , we mean a homeomorphism f : A → B that induces a bijection between A and
B , i.e., for all σ ∈ A , f ( σ ) ∈ B . The reason we have introduced Whitney stratifications and
Whitney stratified sets is the following result, a version of the so-called Thom’s first isotopy
lemma, which generalizes Ehresmann’s Lemma to more general maps.
Theor em 2 §2 2 (Thom’ s first isotopy lemma). [190 ; Ch. II, (5.1.) and (5.2) ] . Let M be a
smooth manifold and (Ω , W ) a locally closed Whitney stratified set of M and let α : M →
Ò k be a smooth map such that
(i) α : Ω → Ò k is proper,
(ii) α | σ : σ → Ò k is surjective, for each stratum σ ∈ W ; and
(iii) α | σ : σ → Ò k is a submersion, for each stratum σ ∈ W .
Then α : Ω → Ò k is a stratified trivial fiber bundle. That is, there exist a Whitney stratified
set ( F , F ) and a stratified homeomorphism h = ( h Ò k , h F ) : (Ω , M ) → ( Ò k × F , { Ò k } × F )
such that α = h Ò k . □
Remark 2 §2 3 . Note that every submersion is an open map and that Ò k is connected. There-
fore to check that α | σ : σ → Ò k is surjective is enough to check that α ( σ ) is closed. Then,
by connectedness, α ( σ ) = Ò k , since α ( σ ) is both open and closed. ¶
Remark 2 §2 4 . As the codomain of α is Ò k , it follows from the proof of [190 ; Ch. II, (5.2) ]
that we have a trivial fiber bundle and not just a locally trivial fiber bundle. The last sentence
follows from noting that the trivial fibration in the statement of [190 ; Ch. II, (5.2) ] is stratified,
see [190 ; Ch. II, (5.1) ] . ¶
Remark 2 §2 5 . We observe that, since h is a stratified homeomorphism, it follows that for all
x , y ∈ Ω , h F ( x ) = h F ( y ) implies that x and y lie in the same stratum of W . ¶

70 Josué Tonelli-Cueto 2 §2
Remark 2 §2 6 . We note that for each x ∈ Ò k , we have that α − 1 ( x )  F. Further, consider
the stratification
W | α − 1 ( x ) := { σ ∩ α − 1 ( x ) | σ ∈ W , σ ∩ α − 1 ( x ) , ∅ } .
Then one can see that W | α − 1 ( x ) is a Whitney stratification of α − 1 ( x ) and that h F gives a
stratified homeomorphism between ( α − 1 ( x ) , W | α − 1 ( x ) ) and ( F , F ) . Because of this, we can
take as ( F , F ) any of the fibers of α . ¶
2 §2 -2 Mather-Thom theory in action: Theorems 1 §2 2 and 1 §3 2
We are now in a position to prove Theorems 1 §2 2 and 1 §3 2 using Thoms’s first isotopy
lemma, although the former can be proven just with Ehresmann’s lemma. We first prove the
simpler Theorem 1 §2 2 and then the harder Theorem 1 §3 2 .
Proof of Theorem 1 §2 2 . Consider the set
Ω := { ( g , x ) ∈ B × Ó n | f ( x ) = 0 }
where B := B W ( f , ∥ f ∥ W κ ( f ) − 1 ) . If Ω is empty, we are done; so we can assume that Ω is
not empty, and we consider the projection
α : Ω → B
( g , x ) 7→ g .
For any compact set K ⊆ H d [ q ] \ Σ d [ q ] , we have that α − 1 ( K ) is a closed subset of K × Ó n and
so compact. Thus α is proper. Further, we can easily see that for g ∈ B, α − 1 ( g ) = Z Ó ( g ) .
Hence, if we show that α is a trivial fibration, we are done, since all fibers would be homotopy
equivalent and thus all of them would have the same homotopy.
To show that α is a trivial fibration, we will apply Thom’s first isotopy lemma (Theo-
rem 2 §2 2 ). Let А : B × Ó n → Ò q be given by А ( f , x ) := f ( x ) . For every ( g , x ) ∈ B × Ó n ,
D ( g , x ) А = ( R 0
x D x g ) : H d [ q ] × T x Ó n → Ò q
where R 0
x is the evaluation map, defined in Proposition 1 §1 6 . For ( g , x ) ∈ Ω , we have
that D ( g , x ) А is surjective, because D x g is so. To see this, note that x ∈ Z Ó ( g ) and that
g < Σ d [ q ] , by the Condition Number Theorem (Corollary 1 §2 10 ). Hence Ω is a smooth
manifold. On this manifold, α is smooth and we consider the trivial Whitney stratification
{ Ω } . The last step for applying Thom’s first isotopy lemma (Theorem 2 §2 2 ) is to show 1) that
α is a submersion and 2) that its image is closed in B, by Remark 2 §2 3 .
1) D ( g , x ) α is the restriction of the projection H d [ q ] × T x Ó n → H d [ q ] to
T ( g , x ) Ω = { ( h , v ) ∈ H d [ q ] × T x Ó n | h ( x ) + D x g v = 0 } ,
and so it is surjective whenever D x g is so. The latter was shown in the above paragraph for
( g , x ) ∈ Ω . Thus α is a submersion.
2) Let { g k } be a sequence in α (Ω) converging to some g ∈ B. Now, due to the axiom
of choice, there is a sequence { x k } in Ó n such that { ( g k , x k ) } is a sequence in Ω . Since

2 §2 Condition and Homology in Semialgebraic Geometry 71
Ó n is compact, we can assume without loss of generality that { x k } is converging to some
x ∈ Ó n , after passing to a subsequence if necessary. Hence g = α ( g , x ) ∈ α (Ω) and we
are done. Therefore α (Ω) is closed in B.
Since the above holds, we can apply Thom’s first isotopy lemma (Theorem 2 §2 2 ) to the
trivial stratification and the proof concludes. □
The above proof didn’t use stratifications. In the next one, we need to be more careful
and so we will need to introduce a new notion of formula.
Definition 2 §2 2 . Let f ∈ H d [ q ] . A strict formula over f is a monotone formula over f that
contains only atoms of the form ( f i = 0) , ( f i > 0) and ( f i < 0) , and a saturated formula over
f is a purely conjunctive strict formula ϕ of the form
ϕ ≡
q
∧
i =1
( f i ∝ i 0)
where ∝ i ∈ { >, < , = } q .
The term “strict” alludes to the fact that no lax inequalities are allowed and the term
“saturated” to the fact that one cannot add more strict atoms to the formula. Saturated
formulas can be encoded by a sign vector, which will be very useful for proofs.
Definition 2 §2 3 . Let f ∈ H d [ q ] and ϕ a saturated formula over ϕ . The sign vector , sgn ( ϕ ) ,
is the vector sgn ( ϕ ) ∈ { − 1 , 0 , +1 } q given by
sgn l ( ϕ ) :=           
+1 , if ∝ l is >
− 1 , if ∝ l is <
0 , if ∝ l is = .
The boundary order on { − 1 , 0 , +1 } q , ≼ , is the partial order defined, for σ , σ ′ ∈ { − 1 , 0 , +1 } q ,
by
σ ≼ σ ′ : ⇔ for all i ∈ [ q ] , σ i = 0 or σ i = σ ′
i . ( 2.10 )
Note that for all x ∈ S ( f , ϕ ) , where ϕ is a saturated formula over f ∈ H d [ q ] , we have
that sgn ( f ( x )) = sgn ( ϕ ) , where sgn ( f ( x )) is just the vector of signs of f ( x ) . In other words,
x ∈ S ( f , ϕ ) iff sgn ( f ( x )) = sgn ( ϕ ) .
The following lemmas are instrumental in the proof of Theorem 1 §3 2 .
Lemma 2 §2 3 . Let f ∈ H d [ q ] . For every Boolean formula Φ over f , there is a unique strict
formula s DNF (Φ) , called strict disjunctive normal form, of the form
s DNF (Φ) ≡ ∨
ξ ∈ Ξ
ϕ ξ
where I is finite and the ϕ ξ are distinct saturated formulas over f such that for all m ≥ 0 and
polynomial tuples g ∈ Ò [ X 0 , . . . , X m ] q ,
W ( g , Φ) = W ( g , s DNF (Φ)) .
In particular, S ( f , Φ) = S ( f , s DNF (Φ)) .

72 Josué Tonelli-Cueto 2 §2
Lemma 2 §2 4 . Let f ∈ H d [ q ] and ϕ and ψ be saturated formulas over f .
(a) If S ( f , ϕ ) ∩ S ( f , ψ ) , ∅ , then sgn ( ϕ ) ≼ sgn ( ψ ) .
(b) If κ ( f ) < ∞ and S ( f , ϕ ) , ∅ , then the following are equivalent:
• S ( f , ϕ ) ⊆ S ( f , ψ ) .
• S ( f , ϕ ) ∩ S ( f , ψ ) , ∅ .
• sgn ( ϕ ) ≼ sgn ( ψ ) .
In particular, when κ ( f ) < ∞ and sgn ( ϕ ) ≼ sgn ( ψ ) , S ( f , ϕ ) , ∅ implies S ( f , ψ ) , ∅ .
(c) If κ ( f ) < ∞ and S ( f , ϕ ) , ∅ , then
S ( f , ϕ ) = S ( f , ϕ ) = ∪ { S ( f , ψ ) | sgn ( ψ ) ≼ sgn ( ψ ) }
where ϕ is the purely conjunctive lax formula obtained substituting ( f i > 0) by ( f i ≥ 0)
and ( f i < 0) by ( f i ≤ 0) .
Proof of Theorem 1 §3 2 . Let B := B W ( f , min i ∥ f i ∥ W κ ( f ) − 1 ) and consider the proper projec-
tion
α : B × Ó n → B
( g , x ) 7→ g .
For each saturated formula ϕ over f , consider the semialgebraic set
Ω( ϕ ) := { ( g , x ) ∈ B × H d [ q ] | x ∈ S ( g , ϕ g ) } .
We can see that
S := { Ω( ϕ ) | ϕ is a saturated formula over f , Ω( ϕ ) , ∅ }
is a locally finite partition of B × H d [ q ] .
Arguing as in the proof of Theorem 1 §2 2 , we can show that the zero set of
А ϕ : B × Ó n → Ò L ( ϕ )
( g , x ) 7→ g L ( ϕ ) ( x )
where L ( ϕ ) := { l ∈ [ q ] | sgn l ( ϕ ) = 0 } is a locally closed submanifold and that the
restriction α | А − 1
ϕ (0) is a submersion. Now, since Ω( ϕ ) = А − 1
ϕ (0) ∩ U ( ϕ ) where
U ( ϕ ) := { ( g , x ) ∈ B × Ó n | for l ∈ [ q ] , f l ( x ) > 0 if sgn l ( ϕ ) = +1
and f l ( x ) < 0 if sgn l ( ϕ ) = − 1 } ,
the same applies to Ω( ϕ ) , i.e., Ω( ϕ ) is a locally closed submanifold such that α | Ω( ϕ ) is
a submersion.

2 §2 Condition and Homology in Semialgebraic Geometry 73
To apply Thom’s first isotopy lemma (Theorem 2 §2 2 ), we need to do two things: 1) show
that S satisfies Whitney’s condition b and therefore is a Whitney stratification, and 2) show
that for each Ω( ϕ ) , α (Ω( ϕ )) is closed in B, by Remark 2 §2 3 .
1) By Lemma 2 §2 4 , we can see that for Ω( ϕ ) ∈ S ,
Ω( ϕ ) = { ( g , x ) ∈ B × Ó n | x ∈ S ( g , ϕ g ) } = ∪ { Ω( ψ ) ∈ S | sgn ( ψ ) ≼ sgn ( ϕ ) } .
This means that we only have to check Whitney’s condition b for ( g , x ) ∈ Ω( ψ ) ∩ Ω( ϕ ) with
sgn ( ψ ) ≼ sgn ( ϕ ) . Let us choose some local chart ( U , u ) around ( g , x ) , in which ( g , x ) is
written as u . We will work in the coordinates of this chart to avoid tedious notation, so that
we write u , v , . . . instead of ( g , x ) , ( h , y ) , . . . . Let { u ℓ } be the sequence in Ω( ψ ) converging
to u and { v ℓ } the sequence in Ω( ϕ ) converging to u .
On the one hand, we can see that
lim
ℓ →∞ T v ℓ Ω( ϕ ) = lim
ℓ →∞ ( v ℓ + ker D v ℓ А ϕ ) = u + ker D u А ϕ .
On the other hand, since u ℓ , v ℓ converges to some line, we can assume without loss of
generality that u ℓ − v ℓ
∥ u ℓ − v ℓ ∥ is convergent, after passing to a subsequence if necessary. We can
see that its limit is the direction vector of the limiting line lim ℓ →∞ u ℓ , v ℓ . Therefore we only
need to show that
lim
ℓ →∞ D u А ϕ ( u ℓ − v ℓ
∥ u ℓ − v ℓ ∥ ) = 0 .
By continuity, this is the same as
lim
ℓ →∞ D v ℓ А ϕ ( u ℓ − v ℓ
∥ u ℓ − v ℓ ∥ ) = 0 .
Now, since А ϕ ( v ℓ ) = 0 , by hypothesis, and А ϕ ( u ℓ ) = 0 , since А ψ ( u ℓ ) = 0 by hypothesis,
we have that
    D v ℓ А ϕ ( u ℓ − v ℓ
∥ u ℓ − v ℓ ∥ )     ≤ 1
2 max
w ∈ [ u ℓ , v ℓ ]     D 2
w А ϕ ( u ℓ − v ℓ
∥ u ℓ − v ℓ ∥ , u ℓ − v ℓ )    
≤ 1
2 max
w ∈ [ u ℓ , v ℓ ]   D 2
w А ϕ   ∥ u ℓ − v ℓ ∥ ,
by Taylor’s theorem. Hence, the desired limit is zero and Whitney’s condition b holds.
2) Take Ω( ϕ ) ∈ S . Let g ∈ B be a limit point of α (Ω( ϕ )) , we only have to show that
S ( g , ϕ g ) , ∅ in order to show that g ∈ α (Ω( ϕ )) . Take any sequence { g k } in α (Ω( ϕ ))
that converges to g . By the axiom of choice, we have that there is a sequence { x k } in Ó n
such that { ( g k , x k ) } lies in Ω( ϕ ) . By compactness of Ó n , we can assume, after taking a
subsequence if necessary, that { x k } converges to some x ∈ Ó n . Now, by the form of the
closure of Ω( ϕ ) , we have that x ∈ S ( g , ϕ g ) and so, by Lemma 2 §2 4 , x ∈ S ( g , ψ g ) for some
saturated formula ψ with sgn ( ψ ) ≼ sgn ( ϕ ) . But then S ( g , ψ g ) , ∅ and so, by Lemma 2 §2 4
again, S ( g , ϕ g ) , ∅ . Thus g ∈ α (Ω( ϕ )) . Since g was arbitrary, this shows that α (Ω( ϕ )) is
closed in B.
At this moment, we can apply Thom’s first isotopy lemma (Theorem 2 §2 2 ) and deduce
that α : B × Ó n → B is a stratified trivial fiber bundle. Note that α was already a trivial

74 Josué Tonelli-Cueto 2 §2
fiber bundle, we need Thom’s first isotopy lemma to guaranteed that it is a stratified trivial
fiber bundle. Following Remark 2 §2 6 above, we note that the induced stratification on Ó n 
α − 1 ( f ) is given by
W f := { S ( f , ϕ ) | ϕ is a saturated formula over f , S ( f , ϕ ) , ∅ }
and so Thom’s first isotopy lemma (Theorem 2 §2 2 ) tells us that there is a continuous map
h : B × Ó n → Ó n such that
( α , h ) : ( B × Ó n , S ) → ( B × Ó n , { B } × { W f } )
is a stratified homeomorphism.
We manipulate this homeomorphism to prove the desired result. Let Φ be an arbitrary
formula over f . By Lemma 2 §2 3 , we can assume without loss of generality that Φ is in the
saturated disjunctive normal form ∨ ξ ∈ Ξ ϕ ξ . Consider the set
Ω(Φ) := ∪
ξ ∈ Ξ
Ω( ϕ ξ )
whose image under ( α , h ) is B × S ( f , Φ) . Since ( α , h ) is a homeomorphism, it restricts to a
homeomorphism
( α ′ , h ′ ) : Ω(Φ) → B × S ( f , Φ)
which induces a homeomorphism between each fiber of α ′ , S ( g , Φ g ) = ( α ′ ) − 1 ( g ) , and
S ( f , Φ) . Hence they have the same homology and we are done. □
Proof of Lemma 2 §2 3 . We apply Morgan’s laws to move negations inwards until they apply
to atoms. Then we just substitute ¬ ( f i = 0) by ( f i , 0) , ¬ ( f i , 0) by ( f i = 0) , ¬ ( f i > 0) by
( f i ≤ 0) , ¬ ( f i ≥ 0) by ( f i < 0) , ¬ ( f i < 0) by ( f i ≥ 0) , and ¬ ( f i ≤ 0) by ( f i < 0) . After this,
we substitute ( f i , 0) by ( f i > 0) ∨ ( f i < 0) , ( f i ≥ 0) by ( f i > 0) ∨ ( f i = 0) , and ( f i ≤ 0) by
( f i < 0) ∨ ( f i = 0) .
Here, we apply the distributive law to take the disjunctions out, until we arrive to a
formula of the form ∨
j ∈ J ∧
k ∈ K j
( f α j ( k ) ∝ j , k 0)
where J is a finite set, { K j } j ∈ J a family of finite sets, { α j : K j → [ q ] } j ∈ J a family of maps
and ∝ ∈ { = , >, < } I × J .
If for some j ∈ J, α j is not injective, then either some factor is repeated inside
∧ k ∈ K j ( f α j ( k ) ∝ j , k 0) or there are two factors that cancel each other and so, we can elimi-
nate it. Therefore, without loss of generality, we can assume that K j ⊆ [ q ] and that α j is the
inclusion map.
If for some j ∈ J, K j , [ q ] , then substitute ∧ k ∈ K j ( f k ∝ j , k 0) by
∨
∝ ′ ∈ { = , >, < } [ q ] \ K j ©  « ∧
k ∈ K j
( f k ∝ j , k 0) ∧ ∧
k ∈ [ q ] \ K j
( f j ∝ ′
k 0) ª ® ¬

2 §2 Condition and Homology in Semialgebraic Geometry 75
where all the summands are saturated. So without loss of generality, K j = [ q ] , and so each
∧ k ∈ [ q ] ( f k ∝ j , k 0) is a saturated formula.
In the last step, we eliminate the repeated summands to arrive to a formula of the desired
form. None of the transformations we have done affects the realization, independently of
which polynomials we substitute in the place of the f i , so it satisfies the desired property.
Further, by substituting f i by X i , we can see that such formula is unique, since
W (( X 1 , . . . , X q ) , s DNF (Φ) ( X 1 , . . ., X q ) ) ∩ { − 1 , 0 , +1 } q = { sgn ( ϕ ξ ) | ξ ∈ Ξ }
which determines uniquely s DNF (Φ) . □
Proof of Lemma 2 §2 4 . (a). When we take a limit point of S ( f , ψ ) , the equalities of the form
f i ( x ) = 0 are preserved and the inequalities f i ( x ) > 0 (resp. f i ( x ) < 0 ) are either preserved
or they turn into equalities of the form f i ( x ) = 0 . This proves the claim.
(b). The first sequence of implications from the first to the third point follows from (a).
For the implication from the third point to the first point, let x ∈ S ( f , ϕ ) and L := { l ∈
[ q ] | sgn l ( ϕ ) = 0 } . Without loss of generality, assume that L = [ k ] . Then by the regularity
inequality (Proposition 1 §3 3 ), D x f L is surjective and so, by the implicit function theorem,
there is an open neighborhood U of x in Ó n and u : U 7→ B (0 , 1) ⊆ Ò n a diffeomorphism,
such that for all i ∈ [ k ] and y ∈ U, u i ( y ) = f i ( y ) . By making U smaller if necessary, we can
assume that for all i ∈ [ q ] and y ∈ U, we have that f i ( y ) > 0 if sgn i ( ϕ ) = sgn i ( ψ ) = +1 ,
and f i ( y ) < 0 if sgn i ( ϕ ) = sgn i ( ψ ) = − 1 . Taking the point,
x t = u − 1 ( u ( x ) + ε
2 ∑
l ∈ L
sgn l ( ψ ) e l )
we can see that for t ∈ (0 , 1) , x t ∈ S ( f , ψ ) , since sgn ( f ( ˜
x )) = sgn ( ψ ) ; and that lim t → 0 x t =
x . Thus x ∈ S ( f , ψ ) and the implication holds.
(c). This follows easily from (b), since the sets S ( f , ϕ ) , with ϕ a saturated formula over
f , cover Ó n . □
2 §2 -3 ( f , λ ) -lartitions and ( f , λ ) -partitions
A polynomial tuple f ∈ H d [ q ] can partition the sphere Ó n in many ways, according to
the values it takes. This motivates the introduction of ( f , λ ) -lartitions and ( f , λ ) -partitions.
The former divide Ó n according to the values that f takes with respect to some finite grid. The
latter consider also the signs that f takes. Depending on the context, one or the other is more
useful: ( f , λ ) -lartitions will appear in the proof of Durfee’ theorem (Theorem 2 §3 2 ) and ( f , λ ) -
partitions in the proof of the Gabrielov-Vorobjov approximation theorem (Theorem 2 §4 2 ).
Definition 2 §2 4 . Let f ∈ H d [ q ] and λ ∈ Ò q × m be a matrix whose entries satisfy λ i , 1 <
· · · < λ i , m , for each i ∈ [ q ] . To each point x ∈ Ó n we associate the following sets:
(J • ) For all k ∈ [ m ] , J • , k ( x ) := { i ∈ [ q ] | f i ( x )/ ∥ f i ∥ W = λ i , k } .
(J ≬ 0) J ≬ , 0 ( x ) := { i ∈ [ q ] | f i ( x )/ ∥ f i ∥ W < λ i , 1 } .
(J ≬ 1) For all k ∈ [ m − 1] , J ≬ , k ( x ) := { i ∈ [ q ] | λ i , k < f i ( x )/ ∥ f i ∥ W < λ i , k +1 } .

76 Josué Tonelli-Cueto 2 §2
(J ≬ 2) J ≬ , m ( x ) := { i ∈ [ q ] | λ i , m < f i ( x )/ ∥ f i ∥ W } .
This defines the ordered partition of [ q ] (in which we allow empty sets):
J ( x ) := ( J ≬ , 0 ( x ) , J • , 1 ( x ) , J ≬ , 1 ( x ) , . . . , J • , m − 1 ( x ) , J ≬ , m − 1 ( x ) , J • , m ( x ) , J ≬ , m ( x )) .
It is clear that the fibers of x 7→ J ( x ) induce an equivalence relation on Ó n . We define the
( f , λ ) -lartition Л f , λ as the set of equivalence classes of this relation.
An ordered partition J := ( J ≬ , 0 , J • , 1 , J ≬ , 1 , . . . , J • , m , J ≬ , m ) of [ q ] defines the set
л J := { x ∈ Ó n | J ( x ) = J } ,
which is an element of Л f , λ , provided it is non-empty.
Remark 2 §2 7 . Less formally, the construction of Л f , λ can be described as follows: the i th
row of the matrix λ ∈ Ò q × ( m +1) defines a partition of Ò into ( −∞ , λ i , 1 ) , { λ i , 1 } , ( λ i , 1 , λ i , 2 ) ,…,
( λ i , m − 1 , λ i , m ) , { λ i , m } , and ( λ i , m , ∞ ) . The product of these partitions of Ò , for i ∈ [ q ] , yields
a partition of Ò q . By taking the preimage of this partition with respect to ˆ
f , we obtain Л f , λ .
So the sets of this partition just indicate us the location of a value f ( x ) = ( y 1 , . . . , y q ) ∈ Ò q
within the discrete grid provided by the matrix λ . ¶
Definition 2 §2 5 . [92 ; Definition 3.6 ] . Let f ∈ H d [ q ] and λ ∈ Ò q × ( m +1) be a matrix whose
entries satisfy 0 = λ i , 0 < λ i , 1 < · · · < λ i , m , for each i ∈ [ q ] . To each point x ∈ Ó n we
associate the following sets:
(I • ) For all 0 ≤ k ≤ m , I • , k ( x ) := { i ∈ [ q ] | | f i ( x ) | / ∥ f i ∥ W = λ i , k } .
(I ≬ 1) For all 0 ≤ k < m , I ≬ , k ( x ) := { i ∈ [ q ] | λ i , k < | f i ( x ) | / ∥ f i ∥ W < λ i , k +1 } .
(I ≬ 2) I ≬ , m ( x ) := { i ∈ [ q ] | λ i , m < | f i ( x ) | / ∥ f i ∥ W } .
This defines the ordered partition of [ q ] (in which we allow empty sets):
I ( x ) := ( I • , 0 ( x ) , I ≬ , 0 ( x ) , I • , 1 ( x ) , I ≬ , 1 ( x ) , . . . , I • , m − 1 ( x ) , I ≬ , m − 1 ( x ) , I • , m ( x ) , I ≬ , m ( x )) .
The point x also determines the tuple of sign conditions σ ( x ) ∈ { − 1 , 0 , +1 } q given by
(S) σ i ( x ) := sgn ( f i ( x )) for i ∈ [ q ] .
It is clear that the fibers of x 7→ ( I ( x ) , σ ( x )) induce an equivalence relation on Ó n . We define
the ( f , λ ) -partition П f , λ as the set of equivalence classes of this relation.
An ordered partition I := ( I • , 0 , I ≬ , 0 , . . . , I • , m , I ≬ , m ) of [ q ] together with a sign vector σ ∈
{ − 1 , 0 , +1 } q defines the set
п I , σ := { x ∈ Ó n | I ( x ) = I , σ ( x ) = σ } ,
which is an element of П f , λ , provided it is non-empty.
Remark 2 §2 8 . Note that ( f , λ ) -partitions are just a particular case of ( f , λ ) -lartitions. They
correspond to the case in which each row of λ is symmetric with respect to the origin.
However, in the symmetric setting where this is needed, ( f , λ ) -partitions are better, because
they encode this symmetry in the sign vector σ . ¶

2 §2 Condition and Homology in Semialgebraic Geometry 77
Remark 2 §2 9 . The term “lartition” is just a language play with the notation that we use.
While we denote the ( f , λ ) -partition by П f , λ , we denote the ( f , λ ) -lartition by Л f , λ . So the
difference in the initial letter reflects the change in the initial letter of their names. ¶
Example 2 §2 5 . Let n = 1 , q = 1 , f = ( XY ) and λ = (0 , √ 3/4) . We can see that, in this
case, Л f , λ has exactly five elements. The zero-dimensional pieces are
{ (1 , 0) , ( − 1 , 0) , (0 , 1) , (0 , − 1) } and { ( 1
2 ,
√ 3
2 ) , ( √ 3
2 , 1
2 ) , ( − 1
2 , − √ 3
2 ) , ( − √ 3
2 , − 1
2 ) } ,
and the one-dimensional pieces
{ ( x , y ) ∈ Ó 2 | 4 x y > √ 3 } , { ( x , y ) ∈ Ó 2 | 0 < 4 x y < √ 3 } and { ( x , y ) ∈ Ó 2 | x y < 0 } .
This is represented in Figure 2 §2 2 . We can see that none of the strata of Л f , λ is connected,
so the strata might be topologically complicated. △
Example 2 §2 6 . [92 ; Example 3.8 ] . Figure 2 §2 3 shows, locally, an example of a ( f , λ ) -
partition on Ó 2 with q = 2 , m = 2 and λ 1 , i = λ 2 , i = λ i . The thick curves correspond
to the zero sets for f 1 and f 2 . The dashed lines are level curves (for both f 1 and f 2 ) with
levels − λ 1 and λ 1 and the dotted curves are the same for the levels − λ 2 and λ 2 . All these
curves partition the picture into 36 two-dimensional open regions, 60 open segments, and
25 points. Each of these 121 regions corresponds to an element in П f , λ . We write down the
details for some of them in Table 2.1 . △
The following two theorems give sufficient conditions on f ∈ H d [ q ] and λ ∈ Ò q × ( m +1)
for the ( f , λ ) -lartition and ( f , λ ) -partition of Ó n to be a Whitney stratification.
Theor em 2 §2 5 . Let f ∈ H d [ q ] with κ ( f ) < ∞ and assume λ ∈ Ò q × m satisfies for i ∈ [ q ] ,
− 1
√ 2 κ ( f )
< λ i , 1 < λ i , 2 < · · · < λ i , m < 1
√ 2 κ ( f )
. ( 2.11 )
Then the ( f , λ ) -lartition Л f , λ is a Whitney stratification of Ó n . Furthermore, under these con-
ditions, the following holds:
(1) The codimension in Ó n of each stratum л J equals ∑ m
k =1 | I • , k | = q − ∑ m
k =0 | I ≬ , k | .
(2) Given л J ∈ Л f , λ and a ∈ J ≬ , k for some k ∈ [ m − 1] , the map
ˆ
f a , J : л J → ( λ a , k , λ a , k +1 )
x 7→ f a ( x )/ ∥ f a ∥ W
is a surjective submersion.
Theor em 2 §2 6 . [92 ; Theorem 3.9 ] Let f ∈ H d [ q ] with κ ( f ) < ∞ and assume λ ∈
Ò q × ( m +1) satisfies for i ∈ [ q ] ,
0 = λ i , 0 < λ i , 1 < · · · < λ i , m < 1
√ 2 κ ( f )
. ( 2.12 )
Then the ( f , λ ) -partition П f , λ is a Whitney stratification of Ó n . Furthermore, under these
conditions, the following holds:

78 Josué Tonelli-Cueto 2 §2
Figure 2 §2 2 : Л f , λ with f = ( XY ) and λ = (0 , √ 3/4)
f 2 = + λ 2
f 2 = + λ 1
f 2 = 0
f 2 = − λ 1
f 2 = − λ 2
f 1 = − λ 2
f 1 = − λ 1
f 1 = 0
f 1 = + λ 1
f 1 = + λ 2
x A
x B
x C
x D
x E
x F
x G
Figure 2 §2 3 : An example of П f , λ (locally) on Ó 2 with q = 2 and m = 2 .
I • , 0 I ≬ , 0 I • , 1 I ≬ , 1 I • , 2 I ≬ , 2 σ
x A ∅ ∅ ∅ ∅ ∅ { 1 , 2 } ( − 1 , − 1)
x B ∅ ∅ ∅ { 1 , 2 } ∅ ∅ ( − 1 , − 1)
x C ∅ ∅ ∅ ∅ { 2 } { 1 } (+1 , − 1)
x D ∅ { 2 } ∅ ∅ ∅ { 1 } ( − 1 , +1)
x E ∅ ∅ ∅ { 1 , 2 } ∅ ∅ ( − 1 , +1)
x F ∅ ∅ { 1 } ∅ { 2 } ∅ (+1 , +1)
x G { 2 } ∅ ∅ ∅ ∅ { 1 } ( − 1 , 0)
Table 2.1 : Some points in Figure 2 §2 3 and their ordered partition and sign vector.

2 §2 Condition and Homology in Semialgebraic Geometry 79
(1) The codimension in Ó n of each stratum п I , σ equals ∑ m
k =0 | I • , k | = q − ∑ m
k =0 | I ≬ , k | .
(2) Given п I , σ ∈ П f , λ and a ∈ I ≬ , k for some k < m , the map
ˆ
f a , I , σ : п I , σ → ( λ a , k , λ a , k +1 )
x 7→ | f a ( x ) | / ∥ f a ∥ W
is a surjective submersion.
Remark 2 §2 10 . Recall that the condition “ a ∈ J ≬ , k for some k ∈ [ m − 1] ” can be less
cryptically written as “ f a ( x )/ ∥ f a ∥ W ∈ ( λ a , k , λ a , k +1 ) for some x ∈ п I , σ and k ∈ [ m − 1] ”, or
simply as “ f a / ∥ f a ∥ W ∈ ( λ a , k , λ a , k +1 ) on п I , σ for some k ∈ [ m − 1] ”. Similarly, the condition
“ a ∈ I ≬ , k for some k < m ” can be rewritten as “ | f a ( x ) | / ∥ f a ∥ W ∈ ( λ a , k , λ a , k +1 ) for some
x ∈ п I , σ and k < m ”, or simply as “ | f a | / ∥ f a ∥ W ∈ ( λ a , k , λ a , k +1 ) on п I , σ for some k < m ”. ¶
The following lemma, which is a simple consequence of the Implicit Function Theorem,
will be instrumental in the proof of the above theorems.
Lemma 2 §2 7 . [91 ; Lemma 4.9 ] . For given f ∈ H d [ q ] put g i := f i / | f i | . Fix x ∈ Ó n , and let
r > 0 be such that √ 2 κ ( f ) r < 1 . We define the index set
S := { i ∈ { 1 , . . . , q } | | g i ( x ) | ≤ r }
and set ¯
u := g S ( x ) ∈ Ò S . Then | S | ≤ n , and there exist an open neighborhood O x of x
in Ó n and ε > 0 with the following properties:
(t1) We have | g i ( y ) | > r for all i < S and all y ∈ O x .
(t2) For all i such that g i ( x ) , 0 , the sign of g i does not change on O x .
(t3) The set Z x := { y ∈ O x | f S ( y ) = f S ( x ) } is a smooth submanifold of Ó n of codi-
mension | S | , and there exists a diffeomorphism h such that the diagram
O x Z x × B ( ¯
u , ε )
B ( ¯
u , ε )
h
g S π B
,
commutes (that is, for every i ∈ S , g i becomes a coordinate projection in the coordi-
nates on O x given by h ).
We will call the pair ( O x , h ) a trivializing chart at x . We can describe a point y ∈ O x
by its trivializing coordinates ( z , u ) ∈ Z x × B ( ¯
u , ε ) , where u = ( u i ) i ∈ S and h ( y )=( z , u ) .
In these coordinates, the normalized polynomial g i = f i / ∥ f i ∥ W , for i ∈ S, takes the form
( z , u ) 7→ u i .
Proof of Theorem 2 §2 5 . In order to show that Л f , λ is a Whitney stratification, we notice that
Л f , λ =
q
∧
i =1
Л f i , λ i = { ∩ q
i =1 л i | п i ∈ Л f i , λ i }

80 Josué Tonelli-Cueto 2 §2
where λ i := ( λ i , 1 , . . . , λ i , m ) is the i th row of λ and Л f i , λ i is the ( f i , λ i ) -partition of Ó n . Thus,
by Proposition 2 §2 1 (I), it is enough to show that each Л f i , λ i is a Whitney stratification and
that Л f 1 , λ 1 , . . . , Л f q , λ q are transversal.
Note that Л f i , λ i consists of open sets of the form f − 1
i ( a , b ) , with ( a , b ) an open interval,
or a hypersurface of the form f − 1
i ( t ) , with t = ∥ f i ∥ W λ i , j for some j . By assumption on λ , this
implies that for such t , | t | < ∥ f i ∥ W /( √ 2 κ ( f )) and hence, by the regularity inequality (Propo-
sition 1 §3 3 ) and the implicit function theorem, all the hypersurfaces are smooth. Whitney’s
condition b is verified in a straightforward way so that we conclude that Л f i , λ i is a Whitney
stratification.
We show now that Л f 1 , λ 1 , . . . , Л f q , λ q are transversal. Let л i ∈ Л f i , λ i , for i ∈ [ q ] , and x ∈
∩ i ≤ q л i . It is easy to check that codim T x л i = 1 if i ∈ J • , k ( x ) and codim T x л i = 0 otherwise.
Therefore, abbreviating J • , ∗ ( x ) := ∪ k J • , k ( x ) , we get ∑ q
i =1 codim T x л i = | J • , ∗ ( x ) | . In
addition, when л i is a hypersurface, we have T x л i = ker D x f i , and thus
q
∩
i =1
T x л i = ∩
i ∈ J • , ∗ ( x )
ker D x f i = ker D x f J • , ∗ ( x ) .
By the regularity inequality (Proposition 1 §3 3 ), the codimension of the right-hand side is
| J • , ∗ ( x ) | . This shows that Л f 1 , λ 1 , . . . , Л f q , λ q are in general position. We conclude that Л f , λ
is a Whitney stratification.
The argument above proves also (1).
We prove part (2) in a standard way. First, we show that ˆ
f a , J is a submersion, i.e., that
its gradient is not tangent to п J . Then we show that ˆ
f a , J is closed. One we have done this,
Remark 2 §2 3 finishes the proof, since closed submersions are surjective when the codomain
is connected.
To show that ˆ
f a , J is a submersion, we fix a point p ∈ п J and take trivializing coordi-
nates around it, using Lemma 2 §2 7 . In these coordinates, using the notation explained after
Lemma 2 §2 7 , п I , σ is an open subset of an affine subspace given by
{ U i = λ i , k (0 < k ≤ m , i ∈ J • , k )
λ i , k ≤ U i ≤ λ i , k +1 (0 ≤ k < m , l ≥ 1 , i ∈ J ≬ , k ) ( 2.13 )
whose tangent space is given by the system
{ U i = 0 (0 ≤ k ≤ m , i ∈ J • , k ) . ( 2.14 )
The map ˆ
f a , J in these coordinates becomes the linear map U a . To check that ˆ
f a , J is a sub-
mersion is then enough to check that U a is not identically zero in the tangent space in these
coordinates. Since a < ∪ k J • , k , U a , this is the case and so ˆ
f a , J is a submersion.
To show that ˆ
f a , J is closed, it is enough to show that for every sequence { x k } in п J , if
{ ˆ
f a , J ( x k ) } has a limit λ ∈ ( λ a , k , λ a , k +1 ) , then there exists x ∈ п J such that ˆ
f a , J ( x ) = λ .
As Ó n is compact, we can assume without loss of generality that { x k } converges to a
point x ′ ∈ п J . By continuity, ˆ
f a , J ( x ′ ) = λ . Passing again to trivializing coordinates and using
Lemma 2 §2 7 , we perturb x ′ to a point x whose components in these coordinates are as

2 §3 Condition and Homology in Semialgebraic Geometry 81
follows:
u i :=           
u ′
i + t if for some k we have i ∈ J ≬ , k and u ′
i = λ i , k
u ′
i − t if for some k we have i ∈ J ≬ , k and u ′
i = λ i , k +1
u ′
i otherwise
with a sufficiently small t > 0 . This new point x evaluates to the same value as x ′ under ˆ
f a , J ,
since u a = u ′
a as u ′
a = λ ∈ ( λ a , k , λ a , k +1 ) by hypothesis; and it belongs to л J . Thus it is the
desired point and we are done. □
Proof of Theorem 2 §2 6 . The proof is analogous to the one of Theorem 2 §2 6 . Moreover, we
can avoid redoing the proof altogether, by noting that for µ ∈ Ò q × (2 m +1) given by
µ i := ( − λ i , m , . . . , − λ i , 1 , 0 , λ i , 1 , . . . , λ i , m )
for i ∈ [ q ] , Л f , µ = П f , λ . □
Proof of Lemma 2 §2 7 . Assume first that S is nonempty. The regularity inequality (Proposi-
tion 1 §3 3 ) implies that D x f S is surjective, since √ 2 κ ( f S ) ∥ f S ( x ) ∥
∥ f S ∥ W < 1 . So clearly | S | ≤ m .
Hence the derivative of the map g S at x is surjective as well. The Implicit Function Theorem
implies the existence of a diffeomorphism h and a neighborhood O x satisfying (t3) with Z x
smooth. By shrinking O x , we can guarantee that properties (t1) and (t2) hold. Finally, the
assertion is easily checked when S is empty. □
2 §3 Durfee’s theorem
A fundamental step in the sampling theory of Chapter 4 that will allow us to construct
the simplicial complex homologically equivalent to the considered closed semialgebraic set is
that we need that inclusions of the form S ( f , t , ϕ ) , → S r ( f , t , ϕ ) , with ϕ a purely conjunctive
lax formula over ( f , t ) , to give isomorphisms in homology.
The following theorem is a consequence of results by Durfee [163] concerning algebraic
neighborhoods of algebraic and semialgebraic sets.
Theor em 2 §3 1 (Durfee’ s theorem). [163 ; §3 ] . Let f ∈ H d [ q ] , t ∈ Ò e and ϕ a purely con-
junctive lax formula over ( f , t ) . Then for all sufficiently small r > 0 , the inclusion S ( f , t , ϕ ) , →
S r ( f , t , ϕ ) is a homotopy equivalence. □
To apply the above statement, we need a quantified version in which the meaning of
“sufficiently small” is quantified. This is handled by the next theorem.
Theor em 2 §3 2 (Quantitative Durfee’ s theorem). [91 ; Proposition 4.6 ] and [92 ; Theo-
rem 4.4 ] . Let f ∈ H d [ q ] , t ∈ ( − T , T ) e and r > 0 be such that √ 2 κ ( f )( r + T ) < 1
and Щ ( t ) > 2 r . Then for every purely conjunctive lax formula ϕ over ( f , t ) , the inclusion
S ( f , t , ϕ ) , → S r ( f , t , ϕ ) is a homotopy equivalence.
We actually will prove a stronger version of this theorem. For f ∈ H d [ q ] , t ∈ ( − T , T ) e ,
ϕ lax formula over ( f , t ) and a vector r ∈ Ò q
> , we define
S r ( f , t , Φ) := Φ Ó n ( ˆ
f − 1
i [ t j − r i , t j + r i ] , ˆ
f − 1
i [ t j − r i , ∞ ) , ˆ
f − 1
i ( −∞ , t j + r i ] | i ∈ [ q ] , j ∈ [ e ] ) .

82 Josué Tonelli-Cueto 2 §3
In a more digestable way, S r ( f , t , Φ) is the semialgebraic set obtained by substituting in Φ
the atoms ( f i = ∥ f i ∥ W t j ) by ( f i ≤ ∥ f i ∥ W ( t j + r i )) ∧ ( f i ≥ ∥ f i ∥ W ( t j − r i )) , ( f i ≥ ∥ f i ∥ W t j )
by ( f i ≥ ∥ f i ∥ W ( t j − r i )) and ( f i ≤ ∥ f i ∥ W t j ) by ( f i ≤ ∥ f i ∥ W ( t j + r i )) , and interpreting the
obtained formula in the obvious way. Note that the the difference between S r ( f , t , Φ) and
S r ( f , t , Φ) , for r > 0 , is that while in the latter we do the relaxation with the same r for all
polynomials, in the former we do this with a different constant for each polynomial.
Theor em 2 §3 3 (Str ong quantitative Durfee’ s theorem). Let f ∈ H d [ q ] , t ∈ ( − T , T ) e
and r ∈ Ò q
> be such that
√ 2 κ ( f )( ∥ r ∥ ∞ + T ) < 1 and Щ ( t ) > 2 ∥ r ∥ ∞ . ( 2.15 )
Then for every purely conjunctive lax formula ϕ over ( f , t ) , the inclusion S ( f , t , ϕ ) , →
S r ( f , t , ϕ ) is a homotopy equivalence.
We move r to ε 1 , with sufficiently small ε > 0 , so that we can apply Durfee’s theorem
(Theorem 2 §3 1 ). However, we don’t carry out this motion all at once, but one component of
r at a time, so that it can be easily handled with Mather-Thom theory.
Remark 2 §3 1 . We note that Theorem 2 §3 2 together with its proof can be extended easily
to cover the case in which Φ is a lax formula, not necessarily purely conjunctive. However,
such an extension would make the proof unnecessarily technical. ¶
2 §3 -1 Moving r to the unknown known case 2
We write r ≤ a r ′ when r a ≤ r ′
a and for i , a , r i = r ′
i . We note that whenever r ≤ a r ′ ,
we have the inclusion
S r ( f , t , ϕ ) ⊆ S r ′ ( f , t , ϕ )
between the generalized algebraic neighborhoods of S ( f , t , ϕ ) . It is enough for us to prove
the following proposition, because with it we can prove Theorem 2 §3 3 .
Pr oposition 2 §3 4 . Let f ∈ H d [ q ] , t ∈ ( − T , T ) e and r , r ′ ∈ Ò q
> be such that ( 2.15 ) holds
for both r and r ′ . If for some a ∈ [ q ] , r ≤ a r ′ , then for every purely conjunctive lax formula ϕ
over ( f , t ) , the inclusion S r ( f , t , ϕ ) , → S r ′ ( f , t , ϕ ) is a homotopy equivalence.
Proof of Theorem 2 §3 3 . Let ε > 0 be such that S ( f , t , ϕ ) , → S ε ( f , t , ϕ ) is a homotopy
equivalence and such that ε 1 ≤ r . This number exists by Durfee’s theorem (Theorem 2 §3 1 ).
Consider now the sequence r (0) , . . . , r ( q ) ∈ (0 , ∞ ) q defined by
r ( k )
i := { r i if i > k
ε if i ≤ k
for i ∈ [ q ] and k ∈ { 0 , . . . , q } . Note that r (0) = r , r ( q ) = ε 1 and that for k ∈ [ q ] ,
r ( k ) ≤ k r ( k − 1) . Hence, by Proposition 2 §3 4 , we have a sequence of inclusions
S ε ( f , t , ϕ ) = S r ( q ) ( f , t , ϕ ) , → S r ( q − 1) ( f , t , ϕ ) , → · · · , → S r (0) ( f , t , ϕ ) = S r ( f , t , ϕ )
where each inclusion is a homotopy equivalence. Thus S ( f , t , ϕ ) , → S r ( f , t , ϕ ) is a homo-
topy equivalence and the proof concludes. □
2 The term “unknown known case” is a punchline which points out that the case stated by the non-explicit
theorem is a known case, but it is unknown as we don’t know when we are in that case explicitly.

2 §3 Condition and Homology in Semialgebraic Geometry 83
2 §3 -2 Moving r one step at a time (Proof of Proposition 2 §3 4 )
We now apply Mather-Thom theory to prove Proposition 2 §3 4 , and with it, the quan-
titative Durfee’s theorem (Theorem 2 §3 2 and 2 §3 3 ). The main idea is to construct a ( f , λ ) -
lartition that is compatible with the considered semialgebraic sets, so that we can apply
Thom’s first isotopy lemma (Theorem 2 §2 2 ).
By Proposition 2 §1 2 , we can assume, without loss of generality, that ϕ is in normal
form. It follows from its proof, that the equality S r ( f , t , ϕ ) = S r ( f , t , NF ( ϕ )) still holds if we
put r in the place of r , as long as the inequality Щ ( t ) > 2 ∥ r ∥ ∞ holds. Thus assume that ϕ
has the form
∧
и ∈ И +
( f и ≥ t α ( и ) ∥ f и ∥ W ) ∧ ∧
и ∈ И −
( f и ≤ t α ( и ) ∥ f и ∥ W )
∧ ∧
и ∈ И 0
(( f и ≥ t l b ( и ) ∥ f и ∥ W ) ∧ ( f и ≤ t u b ( и ) ∥ f и ∥ W ))
with И + , И − , И 0 ⊆ [ q ] pairwise disjoint, α : И + ∪ И − → [ e ] and l b , u b : И 0 → [ e ] such that
for all и ∈ И 0 , t l b ( и ) ≤ t u b ( и ) .
To make notation less tedious, we will assume that ∥ f a ∥ W = 1 without loss of generality,
since κ ( f ) is invariant under scaling. Moreover, for fixed a , we will write r := r a , r ′ := r ′
a
and
С ρ := S r ρ ( f , t , ϕ )
for r ρ := ( r 1 , . . . , r a − 1 , ρ , r a +1 , . . . , r q ) . With these notations, observe that r r = r and r r ′ =
r ′ . And so we only need to show that the inclusion ι : С r → С r ′ is a homotopy equivalence.
Consider now т , т ′ > ∥ r ′ ∥ ∞ such that т < т ′ , √ 2 κ ( f )( T + т ′ ) < 1 and Щ ( t ) > 2 т ′ . We
define λ ∈ Ò q × 4 as follows
λ i =
                                    
( t α ( i ) − т ′ , t α ( i ) − т , t α ( i ) − r i , t α ( i ) ) , if i , a and i ∈ И +
( t α ( a ) − т ′ , t α ( a ) − т , t α ( a ) , t α ( a ) + т ) , if i = a ∈ И +
( t α ( i ) , t α ( i ) + r i , t α ( i ) + т , t α ( i ) + т ′ ) , if i , a and i ∈ И −
( t α ( a ) − т , t α ( a ) , t α ( a ) + т , t α ( a ) + т ′ ) , if i = a ∈ И −
( t l b ( i ) − r i , t l b ( i ) , t l b ( i ) + r i , t l b ( i ) ) , if i , a , i ∈ И 0 and l b ( i ) = u b ( i )
( t l b ( a ) − т , t l b ( a ) , t l b ( a ) + т , t l b ( a ) + т ′ ) , if i = a ∈ И 0 and l b ( a ) = u b ( a )
( t l b ( i ) − r i , t l b ( i ) , t u b ( i ) , t u b ( i ) + r i ) , if i , a , i ∈ И 0 and l b ( i ) , u b ( i )
( t l b ( a ) − т , t l b ( a ) , t u b ( a ) , t u b ( a ) + т ) , if i = a ∈ И 0 and l b ( a ) , u b ( a ) .
The choice of λ is done in a way that the strata of Л f , λ intersect nicely with the С ρ . In orther
words, so that we can prove Proposition 2 §3 5 and 2 §3 6 .
For the sake of simplicity, we have to distinguish four cases: 1) a ∈ И + , 2) a ∈ И − , 3)
a ∈ И 0 with l b ( a ) = u b ( a ) , and 4) a ∈ И 0 with l b ( a ) < u b ( a ) . We will just do the cases
1) and 4), because the case 2) can be easily reduced to the case 1), by changing ( f , t ) to
( − f , − t ) , and the case 3) is analogous to the case 4).

84 Josué Tonelli-Cueto 2 §3
Case 1) a ∈ И +
From now own, we assume that a ∈ И + . The following proposition shows how the
strata of Л f , λ intersect the С ρ .
Pr oposition 2 §3 5 . (1) С т is a union of strata of Л f , λ .
(2) Let ρ , ρ ′ ∈ (0 , т ) . For each л J ∈ Л f , λ such that л J ⊆ С т , the following holds:
(i) л J ∩ С ρ = ∅ iff л J ∩ С ρ ′ = ∅ . In this case, л J ⊆ f − 1
a ( t α ( a ) − т ) .
(ii) л J ∩ С ρ = п J iff л J ∩ С ρ ′ = л J . In this case, л J ⊆ f − 1
a [ t α ( a ) , ∞ ) .
(iii) If ∅ , л J ∩ С ρ ⫋ л J , then л J ⊆ f − 1
a ( t α ( a ) − т , t α ( a ) ) and
л J ∩ С ρ = л J ∩ { x ∈ Ó n | f a ( x ) ≥ t α ( a ) − ρ } .
With this proposition, we can proceed to the proof of Theorem 2 §3 3 in the case in which
a ∈ И + .
Proof of Theorem 2 §3 3 for a ∈ И + . Consider the closed set
Ω := С т ∩ f − 1
a ( t α ( a ) − т , t α ( a ) ) ⊆ f − 1
a ( t α ( a ) − т , t α ( a ) )
and the proper smooth map
ϑ : f − 1
a ( t α ( a ) − т , t α ( a ) ) → (0 , т )
x 7→ t α ( a ) − f a ( x ) .
By Proposition 2 §3 5 , Ω is an union of strata of Л f , λ , particularly of those strata in which
f a takes values in ( t α ( a ) − т , t α ( a ) ) . And so, by Theorem 2 §2 5 and the given assumptions,
the hypothesis of Thom’s first isotopy lemma (Theorem 2 §2 2 ) are satisfied. In particular, this
means that there is F ⊆ Ω and a homeomorphism h := ( ϑ , h F ) : Ω → (0 , т ) × F.
Consider now the linear homotopy
v : [0 , 1] × [0 , т ] → [0 , т ]
( s , y ) 7→                 
y , if y ∈ [0 , r ]
(1 − s ) y + s r , if y ∈ [ r , r ′ ]
y + s ( r − r ′ ) ( 1 − 2 y − r ′
т − r ′ ) , if y ∈ [ r ′ , ( r ′ + т )/2]
y , if y ∈ [( r ′ + т )/2 , т ]
( 2.16 )
that restricts to a continuous retraction of [0 , r ′ ] onto [0 , r ] and that leaves fixed every point
in a neighborhood of { 0 , т } . Using v , we define the map
θ : [0 , т ] × Ω → Ω
( s , x ) 7→ h − 1 ( h F ( x ) , v ( s , ϑ ( x ))) .
This continuous map restricts to a continuous retraction of ϑ − 1 (0 , r ′ ) = С r ′ ∩ f − 1
a ( t α ( a ) −
т , t α ( a ) ) onto ϑ − 1 (0 , r ) = С r ∩ f − 1
a ( t α ( a ) − т , t α ( a ) ) , and it leaves every point in a neighborhood
of the boundary of Ω inside С т fixed. This statement is due to Proposition 2 §3 5 and the fact

2 §3 Condition and Homology in Semialgebraic Geometry 85
that θ respects the strata of Л f , λ , since h is a stratified homeomorphism by Thom’s first
isotopy lemma (Theorem 2 §2 2 ). Because of this last point, it can be extended to a continuous
map
Θ : [0 , т ] × С т → С т
( s , x ) 7→ { ϑ ( s , x ) , if x ∈ Ω
x , otherwise .
By the above paragraph, Θ still restricts to a continuous retraction of С r ′ onto С r . Hence
the inclusion ι : С r → С r ′ is a homotopy equivalence. □
Proof of Proposition 2 §3 5 . Since the ( f , λ ) -lartition Л f , λ classifies points in Ó n according to
the values that f takes with respect to λ , we can see that our choice guarantees trivially all
the claims. We prove only the last claim (2)(iii).
By construction of л J , either л J ⊆ f − 1 ( t α ( a ) , ∞ ) , л J ⊆ f − 1 ( t α ( a ) ) , л J ⊆ f − 1 ( t α ( a ) −
т , t α ( a ) ) or л J ⊆ f − 1 ( t α ( a ) − т ) . The rest of the options are excluded, since л J ⊆ С т . Therefore,
because t α ( a ) − ρ ∈ ( t α ( a ) − т , t α ( a ) ) and ∅ , л J ∩ С ρ ⫋ л J , we must have л J ⊆ f − 1 ( t α ( a ) −
т , t α ( a ) ) .
Since л J ⊆ С т , all defining inequalities of C ρ are satisfied, except at most the inequality
f a ≥ t α ( a ) − ρ . Thus enforcing this inequality is the only difference between л J and л J ∩ С ρ ,
which gives the stated equality. □
Case 4) a ∈ И 0 with l b ( a ) < u b ( a )
From now own, we assume that a ∈ И 0 and that l b ( a ) < u b ( a ) . As in case 1), we
have a proposition relating the strata of Л f , λ and the С ρ . Its proof is analogous to that in
case 1), and because of that, we omit the proof.
Pr oposition 2 §3 6 . (1) С т is a union of strata of Л f , λ .
(2) Let ρ , ρ ′ ∈ (0 , т ) . For each л J ∈ Л f , λ such that л J ⊆ С т , the following holds:
(i) л J ∩ С ρ = ∅ iff л J ∩ С ρ ′ = ∅ . In this case, л J ⊆ f − 1
a ( { t l b ( a ) − т , t u b ( a ) + т } ) .
(ii) л J ∩ С ρ = п J iff л J ∩ С ρ ′ = л J . In this case, л J ⊆ f − 1
a [ t l b ( a ) , t u b ( a ) ] .
(iii) If ∅ , л J ∩ С ρ ⫋ л J , then л J ⊆ f − 1
a ( ( t l b ( a ) − т , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) + т ) ) and
л J ∩ С ρ = л J ∩ { x ∈ Ó n | t l b ( a ) − ρ ≤ f a ( x ) ≤ t u b ( a ) + ρ } .
□
As in case 1), with this proposition, we can proceed to the proof of Theorem 2 §3 3 .
Proof of Theorem 2 §3 3 for a ∈ И 0 with l b ( a ) < u b ( a ) . Consider the closed set
Ω := С т ∩ f − 1
a ( ( t l b ( a ) − т , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) + т ) )
⊆ f − 1
a ( ( t l b ( a ) − т , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) + т ) )

86 Josué Tonelli-Cueto 2 §4
and the proper smooth map
ϑ : f − 1
a ( ( t l b ( a ) − т , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) + т ) ) → (0 , т )
x 7→ { t l b ( a ) − f a ( x ) , if f a ( x ) < t l b ( a )
f a ( x ) − t u b ( a ) , if f a ( x ) > t u b ( a ) .
By Proposition 2 §3 6 , Ω is an union of strata of Л f , λ , particularly of those strata in which
f a takes values in ( t l b ( a ) − т , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) + т ) . And so, by Theorem 2 §2 5 and
the given assumptions, the hypothesis of Thom’s first isotopy lemma (Theorem 2 §2 2 ) are
satisfied. In particular, this means that there is F ⊆ Ω and a homeomorphism h := ( ϑ , h F ) :
Ω → (0 , т ) × F.
With the help of the linear homothopy in ( 2.16 ), we define the map
θ : [0 , т ] × Ω → Ω
( s , x ) 7→ h − 1 ( h F ( x ) , v ( s , ϑ ( x ))) .
This continuous map restricts to a continuous retraction of ϑ − 1 (0 , r ′ ) = С r ′ ∩ f − 1
a (( t l b ( a ) −
r ′ , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) + r ′ )) onto ϑ − 1 (0 , r ) = С r ∩ f − 1
a (( t l b ( a ) − r , t l b ( a ) ) ∪ ( t u b ( a ) , t u b ( a ) +
r )) , and it leaves every point in a neighborhood of the boundary of Ω inside С т fixed. This
statement is due to Proposition 2 §3 6 and the fact that θ respects the strata of Л f , λ , since h
is a stratified homeomorphism by Thom’s first isotopy lemma (Theorem 2 §2 2 ). Because of
this last point, it can be extended to a continuous map
Θ : [0 , т ] × С т → С т
( s , x ) 7→ { ϑ ( s , x ) , if x ∈ Ω
x , otherwise .
By the above paragraph, Θ still restricts to a continuous retraction of С r ′ onto С r . Hence
the inclusion ι : С r → С r ′ is a homotopy equivalence, as desired. □
2 §4 Gabrielov-Vorobjov approximation theorem
The main algorithmic techniques to compute the homology that we have available (see
Chapter 3 ) work for closed sets. Unfortunately, many natural and well-posed semialgebraic
sets are not closed, such as the semialgebraic described by
( ( X − Y , Y ) , ( X − Y = 0 ∧ Y > 0) ∨ ( X − Y > 0 ∧ Y = 0) ) . ( 2.17 )
To circumvent this problem we will rely on a beautiful construction by Gabrielov and Vorobjov
in [186] that produces closed semialgebraic approximations to semialgebraic sets.
The idea of the construction by Gabrielov and Vorobjov [186] is to produce a sequence
of steps, each combining relaxations of closed conditions (equalities and lax inequalities) and
strengthenings of open condition (strict inequalities) of the formula. The sequence is such that
what is missed at one step is covered by next step of relaxations and strengthenings.

2 §4 Condition and Homology in Semialgebraic Geometry 87
Definition 2 §4 1 . [92 ; Definition 2.3 ] Given a monotone formula Φ over f ∈ H d [ q ] and pos-
itive δ and ε , the Gabrielov-Vorobjov ( δ , ε ) -block ГВ δ , ε ( f , Φ) is the spherical semialgebraic
set defined by the following rewriting of Φ ,
f i = 0 ⇝ | f i ( x ) | ≤ ε ∥ f i ∥ W ,
f i , 0 ⇝ ( f i ( x ) ≥ δ ∥ f i ∥ W ) ∨ ( f i ( x ) ≤ − δ ∥ f i ∥ W ) ,
f i > 0 ⇝ f i ( x ) ≥ δ ∥ f i ∥ W ,
f i ≥ 0 ⇝ ( f i ( x ) ≥ δ ∥ f i ∥ W ) ∨ ( | f i ( x ) | ≤ ε ∥ f i ∥ W ) ,
f i < 0 ⇝ f i ( x ) ≤ − δ ∥ f i ∥ W , and
f i ≤ 0 ⇝ ( f i ( x ) ≤ − δ ∥ f i ∥ W ) ∨ ( | f i ( x ) | ≤ ε ∥ f i ∥ W ) .
Given δ , ε ∈ (0 , ∞ ) m , the Gabrielov-Vorobjov ( δ , ε ) -approximation ГВ δ , ε ( f , Φ) (of order m )
of S ( f , Φ) is the spherical semialgebraic set given by
ГВ δ , ε ( f , Φ) :=
m
∪
k =1
ГВ δ k , ε k ( f , Φ) . ( 2.18 )
Remark 2 §4 1 . Note that | f i ( x ) | ≤ ε ∥ f i ∥ W is just an abbreviation of ( f i ( x ) ≤ ε ∥ f i ∥ W ) ∧
( f i ( x ) ≥ − ε ∥ f i ∥ W ) and that both Gabrielov-Vorobjov blocks and Gabrielov-Vorobjov ap-
proximations are compact subsets of Ó n . ¶
Remark 2 §4 2 . The symbol “ГВ” used to denote Gabrielov-Vorobjov blocks and approxima-
tions should not be confused with “ Γ B”, the Greek letter ‘ Γ ’ followed by the Latin letter ‘B’.
“ГВ” comes from the initials in the Cyrillic alphabet of the names of Gabrielov (Габриэлов)
and Vorobjov (Воробьев). ¶
The main result of Gabrielov and Vorobjov is the following one.
Theor em 2 §4 1 (Gabrielov-V orobjov appr oximation theorem). [186 ; Theorem 1.10 ] . Let
f ∈ H d [ q ] , Φ be a monotone formula over f , m ∈ Î , and δ , ε ∈ (0 , ∞ ) m . If
0 < ε 1 ≪ δ 1 ≪ · · · ≪ ε m ≪ δ m ≪ 1 , ( 2.19 )
then, for k ∈ { 0 , . . . , m − 1 } , there are homomorphisms
ϕ k : π k ( ГВ δ , ε ( f , Φ)) → π k ( S ( f , Φ))
and
φ k : H k ( ГВ δ , ε ( f , Φ)) → H k ( S ( f , Φ))
that are isomorphisms for k < m − 1 and epimorphisms when k = m − 1 . □
Remark 2 §4 3 . In the above theorem, the expression 0 < a 1 ≪ · · · ≪ a t ≪ 1 means
that there are functions h k : (0 , 1) t − k → (0 , 1) such that 0 < a k < h k ( a k +1 , . . . , a t )
for all k . Unfortunately, no explicit form for the functions h k is given in [186] , although in
principle one should be able to determine them from the proof there. However, let us note
that the functions h k in ( 2.19 ) do not depend continuously on the coefficients of f for an
arbitrary f . This phenomenon can be seen by taking two orthogonal lines and deforming
them continuously onto the same line. ¶

88 Josué Tonelli-Cueto 2 §4
Remark 2 §4 4 . Homotopy groups (without specifying a base point) are only defined for con-
nected spaces. However, the bijection between π 0 ( ГВ δ , ε ( f , Φ)) and π 0 ( S ( f , Φ)) identifies
the connected components of ГВ δ , ε ( f , Φ) and those of S ( f , Φ) . Therefore we can naturally
interpret ϕ k : π k ( ГВ δ , ε ( f , Φ)) → π k ( S ( f , Φ)) , for k > 0 , as the family of maps
{ ϕ k : π k ( C ) → π k ( ϕ 0 ( C )) | C ∈ π 0 ( ГВ δ , ε ( f , Φ)) } .
The assumption of connectedness in [186 ; Theorem 1.10 ] is only for technical ease of the
exposition. ¶
The proof of Theorem 2 §4 1 goes beyond of what we aim to cover. Because of this we
invite the interested reader to read it in [186] . The following two examples try to illustrate the
theorem.
Example 2 §4 1 . [92 ; Example 2.6 ] . Consider the pair ( f , Φ) in ( 2.17 ). For any pair ( δ , ε ) with
0 < ε < δ the block ГВ δ , ε ( f , Φ) is given by
( | x − y | ≤ ε √ 2 ∧ ( y ≥ δ ) ) ∨ ( | y | ≤ ε ∧ ( x − y ≥ − δ √ 2 ) )
and looks as in Figure 2 §4 4 . It is clear that this block is homotopically equivalent to S ( f , Φ) .
△
Figure 2 §4 4 : The Gabrielov-Vorobjov construction for two open half-lines
Example 2 §4 2 . [92 ; Example 2.7 ] The number m of blocks needed in the Gabrielov-Vorobjov
construction to recover the k th homology group of S ( f , Φ) may reach the bound k + 2 in
Theorem 2 §4 1 . This can be seem in the linear case.
For example, let f = ( X , Y ) and consider
Φ ≡ ( X = 0 ∧ Y = 0) ∨ ( X = 0 ∧ Y > 0) ∨ ( X > 0 ∧ Y = 0) ∨ ( X > 0 ∧ Y > 0)
so that S ( f , Φ) is the non-negative quadrant. Now take any sequence 0 < ε 1 < δ 1 < ε 2 <
δ 2 < ε 3 < δ 3 .
At the left of Figure 2 §4 5 we see in light grey shading the block ГВ δ 1 , ε 1 ( f , Φ) . It is not
connected; not even the 0th homology group is correct. At the center of the figure we see
that same first block with ГВ δ 2 , ε 2 ( f , Φ) superimposed in a darker shade of grey. Now the
union of the first two blocks is connected (so H 0 is correct) but not simply connected: the first
homology group is wrong. We obtain a contractible set, homotopically equivalent to S ( f , Φ) ,
when we add the third block, at the right of the figure, to the union. △
The main theorem of this chapter makes the ≪ in the Gabrielov-Vorobjov approximation
theorem (Theorem 2 §4 1 ) explicit in the case of a well-posed polynomial tuple.

2 §4 Condition and Homology in Semialgebraic Geometry 89
First ГВ block First two ГВ blocks Third ГВ block
Figure 2 §4 5 : The Gabrielov-Vorobjov construction for the positive quadrant
Theor em 2 §4 2 (Quantitative Gabrielov-V orobjov appr oximation theorem). [92 ; The-
orem 2.8 ] . In Theorem 2 §4 1 , condition ( 2.19 ) can be replaced by
0 < ε 1 < δ 1 < · · · < ε m < δ m < 1
√ 2 κ ( f ) ( 2.20 )
when κ ( f ) < ∞ .
Example 2 §4 3 . [92 ; Example 2.9 ] . The simple form of the inequalities in ( 2.20 ) requires
well-posedness, i.e., κ ( f ) < ∞ . To see this, consider f = ( X , Y , X − Y ) and
Φ ≡ (( X = 0) ∧ ( Y > 0)) ∨ (( X − Y = 0) ∧ ( Y > 0)) .
The set S ( f , Φ) consists of two half-lines with a common origin but without this origin. Note
that κ ( f ) = ∞ . Figure 2 §4 6 shows S ( f , Φ) at the left. The center and right parts of the
S ( f , Φ) ГВ 0 . 75 , 0 . 5 ( f , Φ) ГВ 0 . 75 , 0 . 25 ( f , Φ)
Figure 2 §4 6 : The Gabrielov-Vorobjov construction for an ill-posed system
figure exhibit two Gabrielov-Vorobjov Approximations for it with m = 1 but different pairs
( δ , ε ) . The middle part shows that the condition ε < δ is not strong enough to guarantee
the conclusions of Theorem 2 §4 1 for m = 1 . An easy computation shows that, in this case,
we need 0 < ε < δ /2 (as in the right part of the figure). △
The idea of the proof of Theorem 2 §4 2 is simple. Instead of proving it from scratch,
we prove that modifications of ( δ , ε ) do not alter the homotopy of the Gabrielov-Vorobjov
approximations and we use this to reduce to the case in which we can apply Theorem 2 §4 1 .
We first show how we will do transform a pair ( δ , ε ) satisfying ( 2.20 ) into a pair ( δ ′ , ε ′ )
satisfying ( 2.19 ) and then we will show how our basic transformation preserves homotopy.
2 §4 -1 Moving ( δ , ε ) to the unknown known case
We write ( δ , ε ) ≤ D , i ( δ ′ , ε ′ ) when ε = ε ′ , δ j = δ ′
j for j , i , and δ i ≥ δ ′
i . Similarly,
we write ( δ , ε ) ≤ E , i ( δ ′ , ε ′ ) when δ = δ ′ , ε j = ε ′
j for j , i , and ε i ≤ ε ′
i . Note that

90 Josué Tonelli-Cueto 2 §4
the calligraphic index D indicates a difference in a δ (and therefore, in an inequality of the
corresponding Gabrielov-Vorobjov system), while a calligraphic E does so for an ε (and
therefore in an equality). These relations capture the notion of a difference in only one entry
of δ or of ε , respectively. The choice of the inequality in the ε s and the δ s is different. This
is done to ensure that if either ( δ , ε ) ≤ D , i ( δ ′ , ε ′ ) or ( δ , ε ) ≤ E , i ( δ ′ , ε ′ ) , then we have, for
all formulas Φ over f , the inclusion
ГВ δ , ε ( f , Φ) ⊆ ГВ δ ′ , ε ′ ( f , Φ) ( 2.21 )
between the corresponding Gabrielov-Vorobjov approximations. We write
( δ , ε ) D , i
⇝ ( δ ′ , ε ′ )
to denote that
( δ , ε ) ≤ D , i ( δ ′ , ε ′ ) or ( δ ′ , ε ′ ) ≤ D , i ( δ , ε ) .
This notation is consistent with the meaning of updating ( δ , ε ) to ( δ ′ , ε ′ ) by updating (either
increasing or decreasing) only δ i to δ ′
i . We similarly define ( δ , ε ) E , i
⇝ ( δ ′ , ε ′ ) .
The following result states the main property of these rewritings.
Pr oposition 2 §4 3 . [92 ; Proposition 3.1 ] . Let f ∈ H d [ q ] , Φ be a strict formula over f , and
δ , δ ′ , ε , ε ′ ∈ Ò m be such that both ( δ , ε ) and ( δ ′ , ε ′ ) satisfy ( 2.20 ) . If either ( δ , ε ) D , i
⇝
( δ ′ , ε ′ ) or ( δ , ε ) E , i
⇝ ( δ ′ , ε ′ ) , then the corresponding inclusion ( 2.21 ) of Gabrielov-Vorobjov
approximations induces a homotopy equivalence.
Proving Theorem 2 §4 2 from this proposition is easy.
Proof of Theorem 2 §4 2 . By the definition of ≪ , it is clear that there exist at least one ( ˜
δ , ˜
ε )
satisfying both ( 2.19 ) and ( 2.20 ). For any ( δ , ε ) satisfying ( 2.20 ), we can easily construct
a sequence ( δ (0) , ε (0) ) , . . . , ( δ ( ℓ ) , ε ( ℓ ) ) of pairs satisfying ( 2.20 ) such that
1. ( δ (0) , ε (0) ) = ( δ , ε ) ,
2. ( δ ( ℓ ) , ε ( ℓ ) ) = ( ˜
δ , ˜
ε ) , and
3. for each p ∈ { 0 , . . . , ℓ − 1 } , there are k p ∈ { D , E } and i p ∈ { 1 , . . . , m } such that
( δ ( p ) , ε ( p ) ) k p , i p
⇝ ( δ ( p +1) , ε ( p +1) ) .
For such a sequence, the isomorphism types of the homology groups of ГВ δ ( p +1) , ε ( p +1) ( f , Φ)
don’t change at each step as a consequence of Proposition 2 §4 3 . Thus ГВ δ , ε ( f , Φ) has ho-
mology groups isomorphic to those of ГВ ˜
δ , ˜
ε ( f , Φ) . The conclusion now follows from applying
Theorem 2 §4 1 to the latter. □
We next focus on the situations ( δ , ε ) D , i
⇝ ( δ ′ , ε ′ ) and ( δ , ε ) E , i
⇝ ( δ ′ , ε ′ ) . These situa-
tions correspond to replacing δ i in the first one and ε i in the second one by some ζ ∈ ( ε i , δ i ) .
Even though we are updating only one entry in the pair ( δ , ε ) , we have to modify the inequal-
ities associated to several polynomials. Instead of doing this replacement simultaneously in

2 §4 Condition and Homology in Semialgebraic Geometry 91
all the inequalities, we do it by steps, in the inequalities corresponding to a single polynomial
at a time. With this intuition at hand, we introduce the semialgebraic sets below.
Fix f ∈ H d [ q ] , a strict formula Φ over f , positive numbers δ , ε , ζ , t , and a ∈ { 0 , . . . , q } .
We define the following spherical semialgebraic sets:
ГВ D , a
δ , ε , ζ , t ( f , Φ) is obtained from Φ by rewriting
f l = 0 ⇝ | f l ( x ) | ≤ ε ∥ f l ∥ W
f l > 0 ⇝           
f l ( x ) ≥ δ ∥ f l ∥ W if l > a
f a ( x ) ≥ t ∥ f a ∥ W if l = a
f l ( x ) ≥ ζ ∥ f l ∥ W if l < a
f l < 0 ⇝           
f l ( x ) ≤ − δ ∥ f l ∥ W if l > a
f a ( x ) ≤ − t ∥ f a ∥ W if l = a
f l ( x ) ≤ − ζ ∥ f l ∥ W if l < a .
( 2.22 )
ГВ E , a
δ , ε , ζ , t ( f , Φ) is obtained from Φ by rewriting
f l = 0 ⇝           
| f l ( x ) | ≤ ε ∥ f l ∥ W if l > a
| f a ( x ) | ≤ t ∥ f a ∥ W if l = a
| f l ( x ) | ≤ ζ ∥ f l ∥ W if l < a
f l > 0 ⇝ f l ( x ) ≥ δ ∥ f l ∥ W
f l < 0 ⇝ f l ( x ) ≤ − δ ∥ f l ∥ W .
( 2.23 )
Consider now δ , ε ∈ (0 , ∞ ) m , c ∈ { D , E } , i ∈ { 1 , . . . , m } , a ∈ [ q ] and ζ , t > 0 . We define
the intermediate Gabrielov-Vorobjov approximations as the sets
ГВ c , i , a
δ , ε , ζ , t ( f , Φ) := ГВ c , a
δ i , ε i , ζ , t ( f , Φ) ∪ ∪
j , i
ГВ δ j , ε j ( f , Φ) . ( 2.24 )
In particular, we can see ГВ D , i , a
δ , ε , ζ , t ( f , Φ) as the result of having replaced δ i by ζ in all the in-
equalities with polynomials f 1 , . . . , f a − 1 , and being in the process of making the replacement
in those inequalities with f a with the parameter t moving from δ i to ζ .
We now observe that for ζ , t , t ′ > 0 with t ≤ t ′ we have the inclusions
ГВ D , i , a
δ , ε , ζ , t ′ ( f , Φ) ⊆ ГВ D , i , a
δ , ε , ζ , t ( f , Φ) and ГВ E , i , a
δ , ε , ζ , t ′ ( f , Φ) ⊇ ГВ E , i , a
δ , ε , ζ , t ( f , Φ) . ( 2.25 )
The crucial fact to prove Theorem 2 §4 2 is that these inclusions induce homotopy equiva-
lences.
Pr oposition 2 §4 4 . [92 ; Proposition 3.2 ] . Let f ∈ H d [ q ] , Φ be a strict formula, δ , ε ∈ Ò m
satisfying ( 2.20 ) , and let i ∈ [ m ] and a ∈ [ q ] . Then,
(1) For all ζ ∈ ( ε i , δ i ) and ε i < t ≤ t ′ < ε i +1 (where ε m +1 = 1/ √ 2 κ ( f ) by convention),
the inclusion ГВ D , i , a
δ , ε , ζ , t ′ ( f , Φ) ⊆ ГВ D , i , a
δ , ε , ζ , t ( f , Φ) induces a homotopy equivalence.
(2) For all ζ ∈ ( ε i , δ i ) and δ i − 1 < t ≤ t ′ < δ i (where δ 0 = 0 by convention), the inclusion
ГВ E , i , a
δ , ε , ζ , t ′ ( f , Φ) ⊇ ГВ E , i , a
δ , ε , ζ , t ( f , Φ) induces a homotopy equivalence.

92 Josué Tonelli-Cueto 2 §4
Again, Proposition 2 §4 3 easily follows from this result.
Proof of Proposition 2 §4 3 . Assume that ( δ , ε ) ≤ D , i ( δ ′ , ε ′ ) holds. Then δ i ≥ δ ′
i and with-
out loss of generality, δ i > δ ′
i . The following equalities then follow from the definition of
ГВ 1 , i , a
δ , ε , ζ , t ( f , Φ) (we omit the ( f , Φ) in what follows for simplicity):
• ГВ D , i , 1
δ , ε , δ ′
i , δ i = ГВ δ , ε ,
• ГВ D , i , a
δ , ε , δ ′
i , δ ′
i
= ГВ D , i , a +1
δ , ε , δ ′
i , δ i , for all a ∈ [ q − 1] ,
• ГВ D , i , q
δ , ε , δ ′
i , δ ′
i
= ГВ δ ′ , ε = ГВ δ ′ , ε ′ ,
the last one as, by assumption, ε = ε ′ . These equalities yield the following chain
ГВ δ , ε = ГВ D , i , 1
δ , ε , δ ′
i , δ i ⊆ ГВ D , i , 1
δ , ε , δ ′
i , δ ′
i
= ГВ D , i , 2
δ , ε , δ ′
i , δ i ⊆ ГВ D , i , 2
δ , ε , δ ′
i , δ ′
i
= · · · ⊆ ГВ D , i , q
δ , ε , δ ′
i , δ ′
i
= ГВ δ ′ , ε ′ ,
on which all inclusions induce homotopy equivalences by Proposition 2 §4 4 (1). Hence Propo-
sition 2 §4 3 follows in this case.
For the other cases, i.e., when ( δ ′ , ε ′ ) ≤ D , i ( δ , ε ) or when ( δ , ε ) E , i
⇝ ( δ ′ , ε ′ ) , we
proceed analogously. □
2 §4 -2 Moving ( δ , ε ) one step at a time (Proof of Proposition 2 §4 4 )
We have now all the tools needed to prove Proposition 2 §4 4 and with it to finish the
proof of the Quantitative Gabrielov-Vorobjov Theorem 2 §4 2 . We will only prove part (1) of
Proposition 2 §4 4 as part (2) is proven in an analogous way.
We fix f ∈ H d [ q ] , a strict formula Φ over f , tuples δ , ε ∈ (0 , ∞ ) m , an index i ∈ [ m ] ,
a point ζ ∈ ( ε i , δ i ) , points t < t ′ in the interval ( ε i , ε i +1 ) , and an index a ∈ [ q ] , as in the
statement of Proposition 2 §4 4 and satisfying the hypothesis given there. Since a is fixed, we
can assume ∥ f a ∥ W = 1 without loss of generality after scaling f appropriately.
We also choose positive numbers t 0 , t 1 satisfying
ε i < t 0 < t < t ′ < t 1 < ε i +1
and define the matrix λ ∈ Ò q × (2 m +2) whose l th row λ l is given by
λ l := { (0 , ε 1 , δ 1 , . . . , ε i , ζ , δ i , ε i +1 . . . , ε m , δ m ) , if l , a ,
(0 , ε 1 , δ 1 , . . . , ε i , t 0 , t 1 , ε i +1 , . . . , ε m , δ m ) , if l = a .
( 2.26 )
By construction, this λ satisfies ( 2.12 ). We will assume these conventions throughout this
subsection without further mentioning them explicitly. The matrix λ determines the ( f , λ ) -
partition Π f , λ which, as we saw in Theorem 2 §2 6 , is a Whitney stratification of Ó n .
Recall the intermediate Gabrielov-Vorobjov approximations
ГВ τ := ГВ D , i , a
δ , ε , ζ , τ ( f , Φ) ,
defined in ( 2.24 ), for τ ∈ [ t 0 , t 1 ) . These are compact subsets of Ó n .

2 §4 Condition and Homology in Semialgebraic Geometry 93
Proposition 2 §4 4 claims that ι : ГВ t ′ → ГВ t is a homotopy equivalence. The basic
idea for showing this is to apply Theorem 2 §2 2 to the stratification provided by П f , λ . In a first
step towards this goal, we describe how the strata п I , σ of Π f , λ intersect ГВ τ . The findings
are summarized in the proposition below, whose easy but somewhat cumbersome proof is
postponed to § 2 §4 -2 .
Pr oposition 2 §4 5 . [92 ; Proposition 3.11 ] .
(1) ГВ t 0 is a union of strata of П f , λ .
(2) Let τ , τ ′ ∈ ( t 0 , t 1 ) . For each п I , σ ∈ П f , λ such that п I , σ ⊆ ГВ t 0 , the following holds:
(i) п I , σ ∩ ГВ τ = ∅ if and only if п I , σ ∩ ГВ τ ′ = ∅ . In this case, п I , σ ⊆ | f a | − 1 ( t 0 ) .
(ii) п I , σ ∩ ГВ τ = п I , σ if and only if п I , σ ∩ ГВ τ ′ = п I , σ .
(iii) If ∅ , п I , σ ∩ ГВ τ ⫋ п I , σ , then п I , σ ⊆ | f a | − 1 ( t 0 , t 1 ) and
п I , σ ∩ ГВ τ = п I , σ ∩ { x ∈ Ó n | | f a ( x ) | ≥ τ } .
Homotopies pr eserving П f , λ
We are now going to construct the maps and homotopies to show that the inclusion
ι : ГВ t ′ → ГВ t is a homotopy equivalence. For this, we should construct a continuous map
ρ : ГВ t → ГВ t ′ and homotopies between the compositions of these maps and the identity
maps.
A first approach would be to move around the points of ГВ t \ ГВ t ′ and then extend
the maps obtained continuously to the whole space. It is easier though to work in the larger
space ГВ t 0 ∩ | f a | − 1 ( t 0 , t 1 ) , where we can control what happens at the boundary and thus
obtain the continuous extensions.
Consider the open subset M := Ó n \ f − 1
a (0) of Ó n together with the smooth map
M → Ò , s 7→ | f a ( x ) | , as well as the locally closed set
Ω := ГВ t 0 ∩ | f a | − 1 ( t 0 , t 1 ) ⊆ M .
By Proposition 2 §4 5 (1), Ω is the union of certain strata п I , σ of П f , λ , namely of those strata on
which | f a | takes values in ( t 0 , t 1 ) . We note that the restriction of | f a | ,
α : Ω → ( t 0 , t 1 )
x 7→ | f a ( x ) | ,
is a proper map. Indeed, the inverse image α − 1 ( J ) = { x ∈ ГВ t 0 | f a ( x ) ∈ J } of a compact
subset J ⊆ ( t 0 , t 1 ) is a closed subset of the compact set ГВ t 0 and thus compact itself.
By Theorem 2 §2 6 , П f , λ restricts to a Whitney stratification of Ω and the map α satisfies
the hypothesis of Thom’s first isotopy lemma (Theorem 2 §2 2 ). Therefore, there is a subset
F ⊆ Ω and a homeomorphism h : Ω → F × ( t 0 , t 1 ) such that the following diagram commutes
Ω F × ( t 0 , t 1 )
( t 0 , t 1 ) .
h
α π ( t 0 , t 1 )

94 Josué Tonelli-Cueto 2 §4
Moreover, the stratum in which x ∈ Ω lies only depends on h F ( x ) , that is, if h F ( x ) = h F ( y )
then x and y belong to the same stratum of П f , λ . This is so because h is a stratified home-
omorphism by Thom’s first isotopy lemma (Theorem 2 §2 2 ).
Consider the following continuous (piecewise linear) map
υ : [0 , 1] × [ t 0 , t 1 ] → [ t 0 , t 1 ]
( s , y ) 7→                 
y if y ∈ [ t ′ , t 1 ] ,
(1 − s ) y + s t ′ if y ∈ [ t , t ′ ] ,
t 0 + t
2 + ( (1 − s ) + s 2 t ′ − t − t 0
t − t 0 ) ( y − t 0 + t
2 ) if y ∈ [( t 0 + t )/2 , t ] ,
y if y ∈ [ t 0 , ( t 0 + t )/2] .
One easily verifies that this map restricts to a continuous retraction of [ t , t 1 ] onto [ t ′ , t 1 ]
that leaves fixed all points in a neighborhood of { t 0 , t 1 } . With the help of υ , one defines the
continuous map
ψ : [0 , 1] × Ω → Ω
( s , x ) 7→ { x , if α ( x ) < (( t 0 + t )/2 , ( t 1 + t ′ )/2) ,
h − 1 ( h F ( x ) , υ ( s , α ( x ))) , otherwise.
The properties of υ and h imply that this map restricts to a continuous retraction of α − 1 [ t , t 1 )
onto α − 1 [ t ′ , t 1 ) that leaves fixed all points in a neighborhood of the boundary ∂ Ω ∩ ГВ t 0 of
Ω in ГВ t 0 (note ∂ Ω ⊆ | f a | − 1 ( { t 0 , t 1 } ) ).
We also have that ψ ( s , п I , σ ) ⊆ п I , σ for all s ∈ [0 , 1] , provided п I , σ ⊆ Ω . This is so, be-
cause the value h F ( x ) determines the stratum to which x belongs and moreover
h F ( ψ ( s , x )) = h F ( x ) .
Since ψ fixes all points in a neighborhood of ∂ Ω ∩ ГВ t 0 , it can be extended to the
continuous map
Ψ : [0 , 1] × ГВ t 0 → ГВ t 0
( s , x ) 7→ { ψ ( s , x ) , if x ∈ Ω ,
x , otherwise .
As we are extending by the identity, all properties of ψ are inherited by Ψ . In other words,
Ψ restricts to a continuous retraction of ГВ t 0 ∩ | f a | − 1 [ t , ∞ )=( ГВ t 0 \ Ω) ∪ α − 1 [ t , t 1 ) onto
ГВ t 0 ∩ | f a | − 1 [ t ′ , ∞ ) and it preserves the stratification П f , λ , i.e., we have Ψ( s , п I , σ ) ⊆ п I , σ , for
all п I , σ ∈ П f , λ contained in ГВ t 0 and all s ∈ [0 , 1] .
We are now ready to conclude. However, as a warning, we note that Ψ does not give a
continuous retraction of ГВ t onto ГВ t ′ . The reason is that ГВ τ = ГВ t 0 ∩ | f a | − 1 [ τ , ∞ ) generally
does not hold!
Proof of Proposition 2 §4 4 . We first show that for all s ∈ [0 , 1] ,
Ψ( s , п I , σ ∩ ГВ t ) ⊆ п I , σ ∩ ГВ t and Ψ( s , п I , σ ∩ ГВ t ′ ) ⊆ п I , σ ∩ ГВ t ′ .
By Proposition 2 §4 5 (2), there are three possible cases for each of these intersections.
We only focus on the third one, (iii) , since the other two cases are straightforward. In this

2 §4 Condition and Homology in Semialgebraic Geometry 95
case, we have п I , σ ∩ ГВ t = п I , σ ∩ { x | | f a ( x ) | ≥ t } and | f a | ( п I , σ ) ⊆ ( t 0 , t 1 ) . Thus п I , σ ⊆ Ω
and
п I , σ ∩ ГВ t = п I , σ ∩ | f a | − 1 [ t , ∞ ) .
Since this is the case, again by Proposition 2 §4 5 (2), the same happens for t ′ and so
п I , σ ∩ ГВ t ′ = п I , σ ∩ | f a | − 1 [ t ′ , ∞ ) .
Since Ψ gives a deformation retract of ГВ t 0 ∩ α − 1 [ t , t 1 ) onto ГВ t 0 ∩ α − 1 [ t ′ , t 1 ) , it preserves
the stratification П f , λ , and moreover Ψ gives a continuous retraction of п I , σ ∩ | f a | − 1 [ t , ∞ ) =
п I , σ ∩ ( ГВ t 0 ∩ | f a | − 1 [ t , ∞ )) onto п I , σ ∩ | f a | − 1 [ t ′ , ∞ ) . Hence Ψ must preserve п I , σ ∩ ГВ t and
п I , σ ∩ ГВ t ′ and we have shown the claim.
We conclude that Ψ( s , ГВ t ) ⊆ ГВ t and Ψ( s , ГВ t ′ ) ⊆ ГВ t ′ for all s ∈ [0 , 1] .
This allows us to restrict Ψ to obtain continuous maps
Θ : [0 , 1] × ГВ t → ГВ t
( s , x ) 7→ Ψ( s , x )
and Θ ′ : [0 , 1] × ГВ t ′ → ГВ t ′
( s , x ) 7→ Ψ( s , x ) .
Let ρ : ГВ t → ГВ t ′ be the continuous surjection given by
x 7→ Ψ(1 , x ) .
By examining the three cases of Proposition 2 §4 5 (2), we see that ρ is well-defined. Recall
that ι : ГВ t ′ → ГВ t is the inclusion map. By construction, we have
Θ 0 = id ГВ t , Θ 1 = ρ = ι ◦ ρ , Θ ′
0 = id ГВ t ′ and Θ ′
1 = ρ ◦ ι .
Hence, both ( id ГВ t , ι ◦ ρ ) and ( id ГВ t ′ , ρ ◦ ι ) are pairs of homotopic maps. Thus ι induces a
homotopy equivalence as desired. □
Intersecting ГВ with strata (Pr oof of Pr oposition 2 §4 5 )
Arguing as in the proof of Lemma 2 §2 3 , we can assume, without loss of generality, that
Φ is already in saturated normal form, since this does not change any of the ГВ-sets. So we
write
Φ ≡ ∨
ξ ∈ Ξ
ϕ ξ
where each ϕ ξ is saturated.
As we can take out unions in ( 2.24 ), we have
ГВ τ = ГВ D , i , a
δ , ε , ζ , τ ( f , Φ) = ∪
ξ ∈ Ξ ©  « ГВ D , a
δ i , ε i , ζ , τ ( f , ϕ ξ ) ∪ ∪
j , i
ГВ δ j , ε j ( f , ϕ ξ ) ª ® ¬ . ( 2.27 )
Hence it is enough to consider how the different strata intersect with the sets in the right-
hand side. This is done in Lemmas 2 §4 6 , 2 §4 7 , and 2 §4 8 below. We recall that we assume
∥ f a ∥ W = 1 without loss of generality.
The first lemma deals with the ГВ blocks of the form ГВ δ j , ε j ( f , ϕ ξ ) with j , i , the
second lemma with those of the form ГВ D , a
δ i , ε i , ζ , t 0 , and the third lemma with those of the form
ГВ D , a
δ i , ε i , ζ , τ with τ ∈ ( t 0 , t 1 ) . Of these, the third lemma is the most delicate one, as in this case,
the ГВ blocks do not decompose as a union of strata.

96 Josué Tonelli-Cueto 2 §4
Lemma 2 §4 6 . [92 ; Lemma 3.12 ] . Let ϕ be a saturated formula over f , let j , i and put
δ := δ j , ε := ε j . For every п I , σ ∈ П f , λ the following are equivalent:
(0I1) п I , σ ∩ ГВ δ , ε ( f , ϕ ) , ∅ .
(0I2) п I , σ ⊆ ГВ δ , ε ( f , ϕ ) .
(0I3) sgn ( ϕ ) ≼ σ and for all l ∈ [ q ] ,
{ | f l | / ∥ f l ∥ W ≤ ε on п I , σ , if sgn l ( ϕ ) = 0 ,
| f l | / ∥ f l ∥ W ≥ δ on п I , σ , if sgn l ( ϕ ) , 0 .
Proof. The chain of implications from (0I3) to (0I2) to (0I1) follows directly from the definition
of п I , σ . Therefore we only show that (0I1) implies (0I3).
Let x ∈ п I , σ ∩ ГВ δ , ε ( f , ϕ ) . For each l ∈ [ q ] , we distinguish three cases:
+) If sgn l ( ϕ )=1 , then x ∈ ГВ δ , ε ( f , ϕ ) implies f l ( x )/ ∥ f l ∥ W ≥ δ . Therefore, σ l =
sgn ( f l ( x )) = 1 ≽ 1 = sgn l ( ϕ ) and | f l | / ∥ f l ∥ W ≥ δ on п I , σ . The latter because δ
appears in in λ , and so either all x ∈ п I , σ satisfy | f l ( x ) | / ∥ f l ∥ W ≥ δ or none of them
does.
− ) If sgn l ( ϕ ) = − 1 , the argument is analogous to that of the case sgn l ( ϕ ) = 1 .
0) If sgn l ( ϕ ) = 0 , then x ∈ ГВ δ , ε ( f , ϕ ) implies | f l ( x ) | / ∥ f l ∥ W ≤ ε . This, in turn, implies
| f l | / ∥ f l ∥ W ≤ ε on п I , σ , since ε appears in λ , and so either all x ∈ п I , σ satisfy this or
none does. Also 0 ≼ 0 , +1 , − 1 , and so sgn ( f l ( x )) ≼ σ l .
□
Lemma 2 §4 7 . [92 ; Lemma 3.13 ] . Let ϕ be a saturated formula over f . For every п I , σ ∈ П f , λ ,
the following are equivalent:
(1I1) п I , σ ∩ ГВ D , a
δ i , ε i , ζ , t 0 ( f , ϕ ) , ∅ .
(1I2) п I , σ ⊆ ГВ D , a
δ i , ε i , ζ , t 0 ( f , ϕ ) .
(1I3) sgn ( ϕ ) ≼ σ and, for all l ∈ [ q ] ,
              
| f l | / ∥ f l ∥ W ≤ ε i on п I , σ , if sgn l ( ϕ ) = 0 ,
| f l | / ∥ f l ∥ W ≥ δ i on п I , σ , if sgn l ( ϕ ) , 0 and l > a ,
| f l | / ∥ f l ∥ W ≥ t 0 on п I , σ , if sgn l ( ϕ ) , 0 and l = a ,
| f l | / ∥ f l ∥ W ≥ ζ on п I , σ , if sgn l ( ϕ ) , 0 and l < a .
Proof. The proof is analogous to that of Lemma 2 §4 6 , but longer as we must now divide
into cases depending not only on sgn l ( ϕ ) but also on whether l > a , l = a or l < a . □
Lemma 2 §4 8 . [92 ; Lemma 3.14 ] . Let ϕ be a saturated formula over f and s ∈ ( t 0 , t 1 ) . For
every п I , σ ∈ П f , λ the following are equivalent:
(2I1) п I , σ ∩ ГВ D , a
δ i , ε i , ζ , s ( f , ϕ ) , ∅ .

2 §4 Condition and Homology in Semialgebraic Geometry 97
(2I2) sgn ( ϕ ) ≼ σ and for all l ∈ [ q ] ,
              
| f l | / ∥ f l ∥ W ≤ ε i on п I , σ , if sgn l ( ϕ ) = 0 ,
| f l | / ∥ f l ∥ W ≥ δ i on п I , σ , if sgn l ( ϕ ) , 0 and l > a ,
| f l | / ∥ f l ∥ W > t 0 on п I , σ , if sgn l ( ϕ ) , 0 and l = a ,
| f l | / ∥ f l ∥ W ≥ ζ on п I , σ , if sgn l ( ϕ ) , 0 and l < a .
Additionally, if any of the two claims above holds,
п I , σ ∩ ГВ D , a
δ i , ε i , ζ , τ ( f , ϕ ) = { п I , σ ∩ { x ∈ Ó n | | f a ( x ) | ≥ τ } , if | f a | ( t 0 , t 1 ) ⊆ п I , σ ,
п I , σ , otherwise.
( 2.28 )
Proof. The implication from (2I1) to (2I2) is shown in a similar way as those from (0I1) to (0I3)
in Lemma 2 §4 6 and from (1I1) to (1I3) in Lemma 2 §4 7 . We next prove the reverse implication.
Assume then that (2I2) holds. From the conditions there and the definition of both п I , σ
and ГВ D , a
δ i , ε i , ζ , s ( f , ϕ ) , it follows that
п I , σ ∩ ГВ 1 , a
δ i , ε i , ζ , s ( f , ϕ ) = п I , σ ∩ { x ∈ Ó n | | f a ( x ) | ≥ s } . ( 2.29 )
We next divide in cases depending on whether п I , σ ⊆ | f a | − 1 ( t 0 , t 1 ) or not.
⊈ ) If | f a | ( t 0 , t 1 ) ⊈ п I , σ , then | f a | ≥ t 1 on п I , σ , by (2I2), since t 1 is the next value in λ a . This
shows that
п I , σ ∩ { x ∈ Ó n | | f a ( x ) | ≥ s } = п I , σ . ( 2.30 )
As п I , σ is non-empty, (2I1) follows from ( 2.29 ) and ( 2.30 ).
⊆ ) If, instead, | f a | ( t 0 , t 1 ) ⊆ п I , σ then, by Theorem 2 §2 6 (2), the map
п I , σ → ( t 0 , t 1 )
x 7→ | f a ( x ) |
is surjective. Hence п I , σ ∩ { x ∈ Ó n | | f a ( x ) | ≥ τ } is non-empty and (2I1) also follows
in this case.
We have proved ( 2.28 ) in passing. □
Now we finish the proof of Proposition 2 §4 5 with the help of the above three lemmas.
Proof of Proposition 2 §4 5 . Part (1) follows directly from Lemmas 2 §4 6 and 2 §4 7 since these
lemmas guarantee that each set in the right-hand side of ( 2.27 ) is a union of strata.
We now show part (2). Consider the intersections of п I , σ with the decomposition ( 2.27 )
for ГВ τ and ГВ t 0 .
If for some j , i and ξ ∈ Ξ we have п I , σ ∩ ГВ δ j , ε j ( f , ϕ ξ ) , ∅ , then this intersection
equals п I , σ by Lemma 2 §4 6 and all the claims of (2) hold trivially since п I , σ ∩ ГВ δ j , ε j ( f , ϕ ξ )
does not depend on the value of τ .
Assume instead that for all j , i and ξ ∈ Ξ we have п I , σ ∩ ГВ δ j , ε j ( f , ϕ ξ ) = ∅ . Then
п I , σ ∩ ГВ τ = ∪
ξ ∈ Ξ
п I , σ ∩ ГВ D , a
δ i , ε i , ζ , τ ( f , ϕ ξ )

98 Josué Tonelli-Cueto 2 §4
and
п I , σ ∩ ГВ t 0 = ∪
ξ ∈ Ξ
п I , σ ∩ ГВ D , a
δ i , ε i , ζ , t 0 ( f , ϕ ξ ) .
By hypothesis on п I , σ , we have п I , σ ∩ ГВ t 0 = п I , σ , ∅ which implies that there exists
ξ ∈ Ξ such that п I , σ ∩ ГВ D , a
δ i , ε i , ζ , t 0 ( f , ϕ ξ ) , ∅ . Lemma 2 §4 7 then ensures that the conditions
in (1I3) hold true. But these conditions are the same as those in Lemma 2 §4 8 (2l2) except
for l = a , where the inequality is strict in the latter and lax in the former. This means that
п I , σ ∩ ГВ D , i , a
δ , ε , ζ , τ ( f , Φ) = ∅ if and only if | f a | = t 0 on п I , σ . Furthermore, this latter condition is
independent of the particular value of τ . If it holds for τ , then it holds for τ ′ and viceversa.
This proves the first claim of (2).
Arguing as above, we have that п I , σ ∩ ГВ D , i , a
δ , ε , ζ , s ( f , Φ) = п I , σ if and only if | f a | ≥ t 1 on п I , σ .
As this does not depend on the value of τ , we get the second claim of (2).
The third claim of (2) follows directly from the last statement of Lemma 2 §4 8 . □
Further comments
The results in this chapter go back to [88] , Proposition 2 §1 3 ; to [91] , quantitative Dur-
fee’s theorem (Theorem 2 §3 2 ); and to [92] , quantitative Gabrielov-Vorobjov approximation
theorem (Theorem 2 §4 2 ). However, the proofs of some of these results are very different.
The original proof of the Proposition 2 §1 3 in [88] used a continuous version of Smale’s
α -theory (see [88 ; §3.2 and §4.2 ] ), and it only applies to semialgebraic sets in which t =
0 . In contrast to this, our proof used a discontinuous version of the Newton vector field.
Further, we were able to improve the inequality from the original 13 D 3
2 κ ( f ) 2 r < 1 in [88 ;
Theorem 4.19 ] to the current √ 2 κ ( f ) r < 1 , in the case t = 0 .
The proof of Durfee’s theorem (Theorem 2 §3 2 ) differs significantly from that in [91] (in-
cluding the adaptation sketched in [92] ). Instead of doing the continuous retraction in one
step (as done in [91] ), we divide it into different steps. This different proof strategy makes the
proof more similar to that of the Gabrielov-Vorobjov approximation theorem (Theorem 2 §4 2 ),
which helps to introduce the proof of this theorem.
The proof given of the Gabrielov-Vorobjov approximation theorem (Theorem 2 §4 2 ) is
almost identical to that in [92] . Further, there are no significant differences between the two
expositions and, except for some minor changes, the text is the same as that of [92] .
On top of these differences, we have enlarged significantly the exposition of the Mather-
Thom theory done in [91, 92] . Apart from being more systematic, we add some new ex-
amples and theorems (such as Theorem 1 §2 2 and 1 §3 2 ) to illustrate better the applications
and use of this theory. Further, the reader should note in the proofs of this chapter a repeti-
tious style. This style was intentional and it was to emphasize the main ideas and techniques
behind the application of Mather-Thom theory.

99
Pointiller, est le mode d’expression choisi par le peintre qui pose de la couleur sur une toile par
petits points plutôt que de l’étaler à plat.
Paul Signac, D’Eugène Delacroix au néo-impressionnisme
3
Computing the homology of a set
par pointillage
The idea of approximating images by dots is an old one. This variant of approximating
the continuous by the discrete made its first appearance in the mosaics of Antiquity. In
engraving, it was introduced by Giulio Campagnola, Ottavio Leoni and others in the 15th
century, and it was later perfected, under the name of stipple engraving , by Jean Charles
Françoise, William Wynne Ryland and Francesco Bartolozzi in the 18th century [Q13 ; Ch. IX ] .
In painting, it was introduced, with special attention to the colors, by Georges Seurat and Paul
Signac in the 1880s [Q14] , marking the birth of the divisionism (also called, contemptuously
then and popularly now, pointillism ).
Aside from the history of art, Paul Nipkow was the first in conceiving with his Elektrisches
Teleskop in 1884 that an image could be recorded using a finite set of pixels ( Bildpunkte for
him) [Q10] . His design became the germ that, after the contributions of too many inventors
to mention them one by one, gave origin to the screens that inhabit our technological world.
In these screens, the paradoxical illusion of a continuous image made out of discrete dots
shows how well the discrete can approximate the continuous.
At the beginning of the 21st century, the above idea became the foundation of topo-
logical data analysis. In their study of dynamical systems coming from physics, Muldoon,
MacKay, Huke and Broomhead [295] had the idea of extracting topological information
about the attractor of the system from time-series of data coming from experiments. Al-
though this marks the beginning of topological analysis, they were Robins [333] , relying on
work by Robins, Meiss and Bradley [334, 335] , and Edelsbrunner, Letscher and Zomoridan
[168, 168] who laid, respectively, the theoretical and algorithmic foundations of this area.
Nowadays, topological data analysis is an established area of mathematics dealing with
theoretical, computational and applied questions about the topological information that can
be extracted about a set from a finite cloud of points approximating it. In this chapter, we

100 Josué Tonelli-Cueto 3 §1
will expose the ideas and techniques from topological analysis that we will use. Our intention
is not to be complete, but to introduce exactly what we will use later, trying to give some
intuition on the techniques.
First, we introduce the Hausdorff distance, the reach and the Niyogi-Smale-Weinberger
approximation theorem which allow us to guarantee that an approximation is topologically
good; second, we develop some explicit lower bounds 1 on the reach that will allow us to
control the size of the approximation explicitly; and third and last, we introduce the Nerve
theorem and other algebraic topological results that will allow us to go from the approximation
to the homology computationally.
3 §1 Approximation of sets by clouds of points
The fundamental result of this section is the Niyogi-Smale-Weinberger theorem (Theo-
rem 3 §1 8 ) of Niyogi, Smale and Weinberger [300, 301] which gives precise sufficient condi-
tions for when a cloud of points is a good topological approximation of a set in terms of two
numerical quantities: the Hausdorff distance and the reach.
3 §1 -1 Hausdorff distance
The Hausdorff distance is one of the possible distances that one can consider between
sets of a metric space. It is based on the simple idea that two sets X and Y are near if every
point of X is near Y and every point of Y is near of X. Note that this is different from the usual
distance between sets,
dist ( X , Y ) := inf { dist ( x , y ) | x ∈ X , y ∈ Y } ,
for which X and Y are near if a point of X and a point of Y are near.
Definition 3 §1 1 . Let X , Y ⊆ Ò m be non-empty compact subsets, the Hausdorff distance
between X and Y, dist H ( X , Y ) , is the real number given by
dist H ( X , Y ) := max { max { dist ( x , Y ) | x ∈ X } , max { dist ( y , X ) | y ∈ Y } } ( 3.1 )
where dist is the Euclidean distance in Ò m . By convention, we define dist H ( ∅ , X ) = ∞ .
Recall the definition of a (Euclidean) r -neighborhood of a compact set X,
U ( X , r ) := { z | dist ( z , X ) ≤ r } = ∪
x ∈ X
B ( x , r ) . ( 3.2 )
The following proposition is immediate from the above definition. It gives a useful equivalent
formulation of the Hausdorff distance, which helps to develop the intuition about this notion.
Pr oposition 3 §1 1 . Let X , Y ⊆ Ò m be compact subsets. Then the following are equivalent:
• dist H ( X , Y ) ≤ r .
• X ⊆ U ( Y , r ) and Y ⊆ U ( X , r ) . □
1 Unsurprisingly, these bounds are in terms of the condition number.

3 §1 Condition and Homology in Semialgebraic Geometry 101
The following examples should clarify and justify the use of the Hausdorff distance in
our setting.
Example 3 §1 1 . Let X = [0 , 1/2] and Y = [0 , 1] be subsets of Ò , we can see that dist ( X , Y ) =
0 since X ⊆ Y, but dist H ( X , Y ) = 1/2 , since Y ⊆ U ( X , r )=[ − r , 1/2 + r ] if and only if
r ≥ 1/2 . This exemplifies why dist is not a good measure of how near are two sets and that
the Hausdorff distance is able to distinguish a proper subset from a set. △
Example 3 §1 2 . Let X n = 1
k Ú ∩ [0 , 1] = { 0 , 1/ k , 2/ k , . . . , ( k − 2)/ k , ( k − 1)/ k , 1 } and
X = [0 , 1] . We can easily see that dist H ( X k , X ) = 1
2 k which goes to zero as k goes to infinity.
Hence, in the Hausdorff distance, X k converges to X. △
The above example shows a general phenomenon. Recall that an r -net of set X is a
discrete subset N ⊆ X such that for every point x ∈ X, there is a point y ∈ N such that
dist ( x , y ) < r . It is easy to prove the following.
Pr oposition 3 §1 2 . Let X ⊆ Ò m be compact and N ⊂ X be discrete. Then N is an r -net of
X iff dist H ( N , X ) < r . □
In this way, in the Hausdorff metric, the discrete approximations of a set converge to
it. This is the main reason why we use the Hausdorff metric to measure how well a cloud
of points approximates a set, since we expect that an ε -net approximates a set better as ε
goes to zero. Also, it is robust in the sense that it allows N not to be included in X, as long
as it is close to X.
We conclude with the following theorem. On the one hand, it justifies why we speak
about Hausdorff distance or metric; on the other hand, it shows that the Hausdorff distance
is a metric with very good properties. 2
Theor em 3 §1 3 . [360 ; Theorems 7.3.1 and 10.7.2 ] Let S ⊆ Ò m and K ( S ) denote the set
of non-empty compact subsets of S . Then dist H is a metric on K ( S ) . Further, ( S , dist ) is
complete (i.e., S is closed) if and only if ( K ( S ) , dist H ) is complete. □
3 §1 -2 Nearest-point retraction and reach
In a metric space, like Ò m , Urysohn’s lemma is an easy exercise for students as every
closed subset X is just the zero set of the non-negative Lipschitz function dist X : z 7→
dist ( z , X ) . One can see that the neighborhoods U ( X , r ) are just the sublevel sets of this
function. For small r , we might expect that the topologies of U ( X , r ) and of X to be similar,
and this can be made precise in topological terms for arbitrary closed sets [333] . However,
we will concern ourselves with a restricted class of closed subsets for which we can answer
positively the following question: how small should r > 0 should be so that X and U ( X , r )
have the same homotopy type?
Our approach to answer this question goes back to Federer [177] and relies on the
so-called nearest-point retraction.
Definition 3 §1 2 . Let X ⊆ Ò m be closed, the nearest-point retraction is the partial map
π X : Ò m d X defined by
π X ( z ) := arg min { dist ( z , x ) | x ∈ X } . ( 3.3 )
2 We state the result for subsets of Ò m for concreteness, but it holds for general metric spaces.

102 Josué Tonelli-Cueto 3 §1
Figure 3 §1 1 : X, in red, and part of its medial axis, ∆ X , in blue.
The medial axis of X, ∆ X , is the set of those points of Ò m for which π X is not defined.
Note that π X ( z ) is the nearest point in X to z , and that this is well-defined if and only if
this point is unique. With this observation, we can easily see that
∆ X = { z ∈ Ò m | \ x , ˜
x ∈ X : x , ˜
x and dist ( z , x ) = dist ( z , ˜
x ) = dist X ( z ) } , ( 3.4 )
i.e., that ∆ X is the set of those points with two or more distinct nearest points in X.
Example 3 §1 3 . In Figure 3 §1 1 , one can see in red the set X and in blue its medial axis ∆ X .
An observation that should be made in this this picture is that the points of X nearer to ∆ X
are those of higher curvature. △
Pr oposition 3 §1 4 . [177 ; Theorem 4.8(3,4,5) ] . Let X ⊆ Ò m be closed. Then:
(1) The map π x : Ò m \ ∆ X → X is a surjective continuous map.
(2) The map dist X is a C 1 -function on the interior of Ò m \ (∆ X ∪ X ) such that its gradient is
given by
+ X ( z ) := z − π X ( z )
dist X ( z ) . ( 3.5 )
Proof. 1. The surjectivity is obvious, because for all x ∈ X, x is the unique nearest point in
X to x and so π X ( x ) = x .
To prove the continuity, we prove that π X commutes with limits of sequences. Let { z k }
be a sequence of points in Ò m \ ∆ X converging to z ∈ Ò m \ ∆ X . Then,
dist ( π X ( z k ) , z ) ≤ dist ( π X ( z k ) , z k ) + dist ( z k , z )
= dist ( z k , X ) + dist ( z k , z ) ≤ dist ( z , X )+2 dist ( z k , z )
where the first inequality is the triangle inequality, the second one the fact that u 7→ dist ( u , X )
is 1 -Lipschitz, and the equality is by the definition of π X . Since { z k } k is convergent,
{ dist ( z k , z ) } k converges to zero. Therefore, on the one hand, { π X ( z k ) } k is bounded, and,
on the other hand, for every limit point x ∗ of { π X ( z k ) } k ,
dist ( x ∗ , z ) ≤ dist ( z , X ) .

3 §1 Condition and Homology in Semialgebraic Geometry 103
The latter implies that x ∗ = π X ( z ) since π X ( z ) is the unique minimizer of X ∋ x 7→ dist ( x , z ) .
We have just proven that { π X ( z k ) } is a bounded sequence with the unique limit point π X ( z ) .
Thus lim k →∞ π X ( z k ) = π X ( z ) , as we wanted to show.
2. Observe that dist X is a 1 -Lipschitz function. Also, note that for all z ∈ Ò m \ ∆ X and
all λ ∈ [0 , 1] ,
dist X ((1 − λ ) z + λ π X ( z )) = λ dist X ( z ) ,
since, otherwise, there will be a nearer point to z in X distinct from π X ( z ) .
By the above, we obtain that for all sufficiently small t ≥ 0 ,
dist X ( z − t + X ( z )) = dist X ( z ) − t .
Therefore, if dist X is differentiable at z , then
D z dist X ( + X ( z )) = 1 .
But, since ∥ D z dist X ∥ ≤ 1 by the 1 -Lipschitzness of dist X , we must have
D z dist X = + X ( z ) ∗
whenever dist X is differentiable at z .
Finally, since dist X is Lipschitz, dist X is differentiable almost everywhere by Radema-
cher’s theorem [217 ; Theorem 3.1 ] . But then, by the Fundamental Theorem of Calculus for
Lipschitz functions [217 ; Theorem 3.3 ] we have that dist X is differentiable and that + X is its
gradient on Ò M \ ( X ∪ ∆ X ) , since + X is continuous. □
This means that as long as we are away from ∆ X , the nearest-point retraction π X is a
good map and it points to the direction we must take to decrease the distance to X the
fastest possible way.
Reach
To measure how far from the set the nearest point retraction is defined, one considers
the notion of reach.
Definition 3 §1 3 . [177] . Let X ⊆ Ò m be a closed set. The local reach of X at x ∈ X, τ ( X , x ) ,
is the non-negative quantity
τ ( X , x ) := dist ( x , ∆ X ) ( 3.6 )
and the reach (or local feature size ) of X, τ ( X ) , is the non-negative quantity
τ ( X ) := inf
x ∈ X τ ( X , x ) = dist ( X , ∆ X ) . ( 3.7 )
The reach is precisely the quantity we were looking for.
Pr oposition 3 §1 5 . [88 ; Proposition 2.2 ] . Let X ⊆ Ò m be a closed subset such that τ ( X ) > 0 .
Then for all r ∈ (0 , τ ( X )) ,
H X : [0 , 1] × U ( X , r ) → U ( X , r )
( t , z ) 7→ (1 − t ) z + t π X ( z )
is a continuous retraction of U ( X , r ) onto X . In particular, for all r ∈ (0 , τ ( X )) , X , → U ( X , r )
is an homotopy equivalence.

104 Josué Tonelli-Cueto 3 §1
(a) Two intersecting lines (b) Two tangent circles
Figure 3 §1 2 : Sets with reach zero, in red, and their medial axis in blue
Proof. Since dist ( H X ( t , z ) , π X ( z )) ≤ (1 − t ) dist ( z , π X ( z )) = (1 − t ) dist X ( z ) = (1 − t ) r ≤ r ,
H X is well-defined. The continuity of H X follows from Proposition 3 §1 4 and that is a retraction
from the fact that for all x ∈ X, π X ( x ) = X. □
Observation 3 §1 1 . Note that an alternative way to define H X for ( t , z ) with z < X is by
H X ( t , z ) = z − t dist X ( z ) + X ( z ) .
Since + X is the gradient of dist X , this shows that we have just done gradient descent. ¶
There are many sets with positive reach, among them the ones with maximum reach,
i.e., reach equal to infinity, being the convex closed sets. However, let us note that not every
set has positive reach as the following two examples shows.
Example 3 §1 4 . Consider the set which consist in two lines with an small angle of θ between
them. In Figure 3 §1 2 a, one can see the union of the lines, X, in red and the medial axis, ∆ X ,
in blue. In this case, We observe that ∆ X is not closed, since it misses the intersection point
of the lines, and that τ ( X ) = 0 . △
Example 3 §1 5 . Consider the set consisting on the union of two tangent circles. As in the
previous example, in Figure 3 §1 2 b, the set X in red and the medial axis ∆ X in blue. Similarly
to the previous example, ∆ X is not closed, because it misses the intersection point of the
circles, and τ ( X ) = 0 . △
Normal vector and the r each along a dir ection
How far can we leave X from x in the direction given by a vector u before the fastest
path to return to X is the reverse of the path traversed til then? This motivates the following
definition, which is a directional version of the local reach at a point.
Definition 3 §1 4 . [177] . Let X ⊆ Ò m be a compact set, x ∈ X and u ∈ Ó m − 1 . The local
reach along u of X at x , τ ( X , x ; u ) , is the non-negative quantity defined by
τ ( X , x ; u ) := sup { t ≥ 0 | dist ( x + t u , X ) = t } . ( 3.8 )
The importance of the above notion is that it will allow us to work in a easier way with
the reach. The following theorem gives the best way of proving lower bounds.
Theor em 3 §1 6 . [88 ; Lemma 2.5 ] and [177 ; Theorem 4.8(6) ] . Let X ⊆ Ò m be closed, x ∈ X
and u ∈ Ó n . Then

3 §1 Condition and Homology in Semialgebraic Geometry 105
(1) τ ( X , x ; u ) = sup { t ≥ 0 | x + t u < ∆ X and π X ( x + t u ) = x } .
(2) If 0 < τ ( X , x ; u ) < ∞ , then x + τ ( X , x ; u ) u ∈ ∆ X .
(3) If τ ( X , x ; u ) > 0 , then τ ( X , x ; u ) ≥ τ ( X , x ) .
Proof. (1) Given that x + t u < ∆ X , dist ( x + t u , X ) = t and π X ( x + t u ) = x are equivalent.
This shows that
τ ( X , x ; u ) ≥ sup { t ≥ 0 | x + t u < ∆ X and π X ( x + t u ) = x } .
To prove the equality, assume that it does not hold. In this case, there is t 0 in between these
two quantities. For this t 0 , on the one hand, x + t 0 u ∈ ∆ X and dist X ( x + t 0 u ) = t 0 ; on the
other hand, there is ˜
x ∈ X different from x such that dist ( x + t 0 u , ˜
x ) = t 0 .
Let Y = { x , ˜
x } . Then, by elementary geometry, H := ∆ Y is a hyperplane that passes
through x + t 0 u and whose complement is the union of two open half-spaces: U comprising
those points nearer to x and ˜
U comprising those nearer to ˜
x . Clearly, [ x , x + t 0 u ) ⊆ U.
Therefore, for t > t 0 sufficiently near t 0 , t < τ ( X , x ; u ) and
dist ( x + t u , X ) ≤ dist ( x + t u , ˜
x ) < dist ( x + t u , x ) = t ,
since x + t u ∈ ˜
U; and so
dist ( x + τ ( X , x ; u ) u , X ) ≤ τ ( X , x ; u ) − t + dist X ( x + t u ) < τ ( X , x ; u ) .
This contradicts the definition of τ ( X , x ; u ) . Hence the equality must hold.
(2) Let z 0 = x + τ ( X , x ; u ) u . Assume that z 0 < ∆ X . Then, in a neighborhood of z 0 ,
the continuous vector field + X of Proposition 3 §1 4 is defined. By the Cauchy-Peano theo-
rem [180 ; Teorema 2.2a ] , { α ′ ( t ) = + X ( α ( t ))
α (0) = z 0
has at least one local solution α : ( − δ , δ ) → Ò m for some δ > 0 .
Now, by construction of α ,
D α ( t ) dist X ( α ′ ( t )) = D α ( t ) dist X ( + X ( α ( t ))) = 1
where the last claim follows from Proposition 3 §1 4 (2). Therefore, by the chain rule,
( dist X ◦ α ) ′ ( t ) = 1 = ∥ α ′ ( t ) ∥ ,
where the last inequality follows from ∥ + X ∥ = 1 .
By the above paragraph,
∫ t 1
t 0 ∥ α ′ ( s ) ∥ d s = ∫ t 1
t 0
( dist X ◦ α ) ′ ( t ) d s = dist X ( α ( t 1 )) − dist X ( α ( t 0 )) ≤ dist ( α ( t 0 ) , α ( t 1 )) .
Therefore
length α [ t 0 , t 1 ] ≤ dist ( α ( t 0 ) , α ( t 1 ))

106 Josué Tonelli-Cueto 3 §1
where length α [ t 0 , t 1 ] is the length of α between t 0 and t 1 . This means that α is a geodesic
in Ò m with constant unit speed and so it is of the form
α ( s ) = z 0 + s u .
But then for t > τ ( X , x ; u ) sufficiently near to τ ( X , x ; u ) ,
+ X ( x + t u ) = α ′ ( t − τ ( X , x ; u )) = u
and so π X ( x + t u ) = x contradicting the equality in (1).
(3) We have that
τ ( X , x ; u ) = dist ( x , x + τ ( X , x ; u ) u ) ≥ dist ( x , ∆ X ) = dist ( x , ∆ X ) = τ ( X , x )
where the first equality comes from the definition of τ ( X , x ; u ) , and the first inequality from
(2). □
We finish this section with the following proposition which shows for which vectors u we
can guarantee that τ ( X , x ; u ) > 0 when X is a locally closed submanifold around x . Given a
smooth submanifold M ⊆ Ò m and x ∈ M , recall that a normal vector of M at x is a vector
u ∈ T x Ò m such that u is orthogonal to T x M . We denote the vector subspace of normal
vectors of M at x by N x M := T x M ⊥ .
Pr oposition 3 §1 7 . Let M ⊆ Ò m be closed and x ∈ X be such that M is a regular manifold
around x . Then for every u ∈ Ó m − 1 , τ ( X , x ; u ) > 0 iff u ∈ N x M .
Proof. By the Constant Rank Theorem [275 ; Theorem 4.12 ] , there is an open neighborhood
B ( x , ε ) of x and a smooth map f : Ò n → Ò q such that f − 1 (0) ∩ B ( x , ε ) = M ∩ B ( x , ε )
and such that its tangent map of f at every point z ∈ B ( z , ε ) , D z f , is surjective. In this
neighborhood, let
P f , z := É − D z f † D z f
be the orthogonal projection onto ker D z f , which for z ∈ M ∩ B ( x , ε ) is just the tangent
space of M , T z M . Note that z 7→ P f , z is differentiable and that the values of its derivative
when evaluated at a vector are matrices, of which we will consider the spectral norm.
Consider the following minimization problem
arg min { dist ( x + t u , y ) | y ∈ M }
which for t < ε /2 has its solution in M ∩ B ( x , ε ) . Therefore, for such t , by the Lagrange
multipliers theorem [275 ; Exercise 11-11 ] , we have that for any local minimizer in y ∈ M ∩
B ( x , ε ) , x + t u − y is orthogonal to T y M or, equivalently,
P f , y ( x + t u − y ) = 0 .
In the special case in which y = x , this gives P f , y u = 0 and so u ∈ N x M showing one of
the implications.
For the other implication, assume that u ∈ N x M and that there is another local mini-
mizer y ∈ M ∩ B ( x , ε ) different from x . Then
y − x + D y f † f ( y ) + D y f † D y f ( x − y ) − D y f † f ( x ) = P f , y ( y − x ) = t P f , y u = t ( P f , y − P f , x ) u

3 §1 Condition and Homology in Semialgebraic Geometry 107
since f ( x ) = f ( y ) = 0 and P f , x u = 0 . On the one hand,
∥ t ( P f , y − P f , x ) u ∥ ≤ t ∥ P f , y − P f , x ∥ ≤ t max
z ∈ [ x , y ]     D z P f ( y − x
∥ y − x ∥ )     ∥ y − x ∥ ,
where the last inequality follows from Taylor’s theorem with remainder; and, on the other
hand,
∥ y − x + D y f † f ( y ) + D y f † D y f ( x − y ) − D y f † f ( x ) ∥
≥ ( 1 − ∥ D y f † f ( y ) + D y f † D y f ( x − y ) − D y f † f ( x ) ∥
∥ x − y ∥ ) ∥ x − y ∥
≥ ( 1 − 1
2 max
z ∈ [ x , y ] ∥ D y f † D 2
z f ∥ ∥ x − y ∥ ) ∥ x − y ∥ ,
because ∥ D y f † f ( y ) + D y f † D y f ( x − y ) − D y f † f ( x ) ∥ ≤ 1
2 max z ∈ [ x , y ] ∥ D y f † D 2
z f ∥ ∥ x − y ∥ 2
by the Taylor theorem with remainder.
Combining these two inequalities and cancelling ∥ y − x ∥ , we obtain
t max
z ∈ [ x , y ]     D z P f ( y − x
∥ y − x ∥ )     ≥ ( 1 − 1
2 max
z ∈ [ x , y ] ∥ D y f † D 2
z f ∥ ∥ x − y ∥ ) . ( 3.9 )
By continuity of all terms involved, there are δ > 0 and t 0 > 0 such that for all y such that
∥ y − x ∥ < δ ,
max
z ∈ [ x , y ]     D z P f ( y − x
∥ y − x ∥ )     < t − 1
0
and ( 1 − 1
2 max
z ∈ [ x , y ] ∥ D y f † D 2
z f ∥ ∥ y − x ∥ ) > 1
2 .
This means that there can be another local minimizer y in the considered region, only if
t > t 0 /2 . Hence x is the unique local minimizer for t sufficiently small, and so τ ( X , x ; u ) > 0
as desired. □
3 §1 -3 Niyogi-Smale-Weinberger approximation theorem
The Niyogi-Smale-Weinberger approximation theorem gives a sufficient condition for a
cloud of points to approximate the homotopy type of a set. The original versions [300, 301]
had more complicated inequalities and proofs than the version given here [88 ; Theorem 2.8 ]
(cf. [109 ; Theorem 4.6 ] ).
Theor em 3 §1 8 (Niyogi-Smale-W einber ger approximation theor em). [88 ; Theorem 2.8 ] .
Let X , X ⊆ Ò m be compact sets. Assume that τ ( X ) > 0 . Then for all ε > 0 such that
3 dist H ( X , X ) < ε < 1
2 τ ( X ) , ( 3.10 )
X is a deformation retract of U ( X , ε ) . In particular, X , → U ( X , ε ) is an homotopy equiva-
lence.

108 Josué Tonelli-Cueto 3 §1
Proof of Theorem 3 §1 8 . Note that
U ( X , ε ) ⊆ U ( U ( X , dist H ( X , X )) , ε ) ⊆ U ( X , dist H ( X , X ) + ε ) ⊆ Ò m \ ∆ X ,
since, by assumption, dist H ( X , X ) + ε < τ ( X ) . Therefore H x of Proposition 3 §1 5 is well-
defined for all ( t , z ) ∈ [0 , 1] × U ( X , ε ) . We will show that H X gives a continuous retraction
of U ( X , ε ) onto X. For this, by Proposition 3 §1 5 , we only have to show that for all ( t , z ) ∈
[0 , 1] × U ( X , ε ) , H X ( t , z ) ∈ U ( X , ε ) . By the way that H X is constructed, we have to show
that for all z ∈ U ( X , ε ) ,
[ z , π X ( z )] ⊆ U ( X , ε ) .
Let z ∈ U ( X , ε ) , 𝕫 := π X ( z ) and x ∈ X be such that dist ( x , z ) ≤ ε . If dist ( 𝕫 , x ) ≤ ε ,
then [ z , 𝕫 ] ⊆ B ( x , ε ) and we are done. Therefore assume dist ( 𝕫 , x ) > ε from now on. In
this way, let ˜
z ∈ [ z , 𝕫 ] be the nearest point to 𝕫 such that dist ( ˜
z , x ) = ε . We only have to
show then that [ ˜
z , 𝕫 ] ⊆ U ( X , ε ) . In fact, we will show something stronger, namely, that it is
contained in just one of the balls that constitute U ( x , ε ) .
Let
u := z − 𝕫
∥ z − 𝕫 ∥ = ˜
z − 𝕫
∥ ˜
z − 𝕫 ∥ ,
then, since z = 𝕫 + t u for some t > 0 and π X ( z ) = 𝕫 , we have τ ( X , 𝕫 ; u ) > 0 . Hence, by
Theorem 3 §1 6 (3),
τ ( X , 𝕫 ; u ) ≥ τ ( X ) .
Consider now, r := 1
3 ε . On the one hand, 6 r < τ ( X ) , therefore τ ( X , 𝕫 ; u ) > 6 r and we can
construct z := 𝕫 + 6 r u such that π X ( z ) = 𝕫 . On the other hand, dist H ( X , X ) < r , thus there
is ˜
x ∈ X such that dist ( ˜
x , 𝕫 ) < r .
We will be done once we show that dist ( ˜
x , ˜
z ) ≤ ε . We note now that
dist ( ˜
y , ˜
z ) ≤ dist ( ˜
y , 𝕫 )+ dist ( 𝕫 , ˜
z ) ≤ r + dist ( 𝕫 , ˜
z ) = r + dist ( z , 𝕫 ) − dist ( z , ˜
z ) = 7 r − dist ( z , ˜
z ) ,
where the middle equality follows from the fact that 𝕫 , ˜
z and z are collinear with ˜
z in the
middle. Because of this, it is enough to show
4 r ≤ dist ( z , ˜
z ) ,
i.e., that ˜
z is “far” from z .
To end the proof consider the triangle x ˜
z z whose vertices are x , ˜
z and z . Now, the
angle Θ at ˜
z of x ˜
z z is the same angle at ˜
z of the triangle x ˜
z ˜
z ′ , where ˜
z ′ is the point in [ z , z ]
such that dist ( x , ˜
z ′ ) = ε . This last triangle is isosceles, with the considered angle being a
base angle. Thus Θ is acute. We note that in the degenerate situation, Θ = π /2 and the
argument below still applies.
Since x is “near” X and z “far” from X, we have that
dist ( x , z ) ≥ dist X ( z ) − dist X ( x ) ≥ 6 r − r = 5 r
where we have use that dist X ( x ) ≤ dist H ( X , X ) < r . Finally, by the cosine theorem applied
at Θ , we have
dist ( x , z ) 2 = dist ( ˜
z , z ) 2 + dist ( ˜
z , x ) 2 − 2 dist ( ˜
z , z ) dist ( ˜
z , x ) cos Θ ≤ dist ( ˜
z , z ) 2 + dist ( ˜
z , x ) 2

3 §1 Condition and Homology in Semialgebraic Geometry 109
since Θ is acute. From here, we have
dist ( ˜
z , z ) ≥ √ dist ( x , z ) 2 − dist ( ˜
z , x ) 2 ≥ √ 25 r 2 − 9 r 2 ≥ 4 r .
This concludes the proof. □
Remark 3 §1 2 . In case the reader does not like proofs à la Peano , i.e., without drawings [Q12] ,
E can see a diagram of the proof in Figure 3 §1 3 . In this diagram, we have not only repre-
sented the relations of proximity, but also the metric relations concerning the points involved
in the proof. ¶
𝕫𝕫
˜
z
z
z
x
˜
x
X
6 r
r
r
ε
ε
Figure 3 §1 3 : Diagram of the proof of Theorem 3 §1 8
Remark 3 §1 3 . A version of the above theorem (Theorem 3 §1 8 ), but with worse constants can
be obtained as a corollary of the Chazal-Cohen-Steiner-Lieutier approximation theorem [109 ;
Theorem 4.6 ] . The proof there is more involved, but it is so because their theorem applies
to a more general setting. They do this by using the notions of µ -reach, τ µ , and weak reach,
τ w , introduced by Chazal and Lieutier [110, 111] .
Let us briefly recall these notions. Let X ⊆ � n be compact and z ∈ � m . Then define
Γ X ( z ) := { x ∈ X | dist ( z , x ) = dist ( z , X ) } . Let Θ X ( z ) ∈ � n and R X ( z ) ∈ [0 , ∞ ) to be such
that B (Θ X ( z ) , R X ( z )) is the (unique) smallest closed ball containing Γ X ( z ) . Finally, consider
the vector field
� X ( z ) := z − Θ X ( z )
dist X ( z ) ,
which is well-defined whenever z  Θ X ( z ) . One can see that for z  � m \ ( X ∪ ∆ X )
this agrees with the one defined in Proposition 3 §1 4 . However, constructing a flow for this
vector field requires technical tools, since it is not continuous in general, and because the
continuous retraction it gives is only between neighborhoods. In this setting, the µ -reach (for
µ ∈ (0 , 1] ) and weak reach are given by
τ µ ( X ) := dist ( X , { z ∈ � m | ∥ � X ( z ) ∥ < µ } and τ µ ( X ) := dist ( X , { z ∈ � m | ∥ � X ( z ) ∥ = 0 } .

110 Josué Tonelli-Cueto 3 §2
Note that the reach is just the 1 -reach.
One can see that, in Example 3 §1 4 , τ µ ( X ) = ∞ if µ ≤ 1 − cos θ
2
√ ( 1 − cos θ
2 ) 2 + tan 2 θ
2
where θ is
the acute angle between the red lines and τ µ ( X ) = 0 otherwise. However, in Example 3 §1 5 ,
τ w ( X ) > 0 , but τ µ ( X )=0 for all µ > 0 . In general, one can show that for every closed
semialgebraic set, τ w ( X ) > 0 [111 ; Proposition 3.6 ] , but one should observe that in order
to make this computational one needs explicit lower bounds. It is not clear how to extend
some of the results that only work for the µ -reach, or how to give bounds for the µ for which
the µ -reach is positive. We discuss this problem in Chapter 5 . ¶
3 §2 Reach: Explicit lower bounds
In this section, we provide lower bounds for the reach of an intersection in terms of the
reach of the intersection of the boundaries, a lower bound of the local reach of an analytic
set in terms of Smale’s gamma, and a lower bound for the class of spherical semialgebraic
sets we will be working with.
3 §2 -1 Reach of intersections
The main theorem we will prove is the following one.
Theor em 3 §2 1 . [88 ; Corollary 2.6 ] . Let { X i } i ∈ I be a finite family of closed subsets of Ò m
and Y ⊆ Ò m another closed subset. Then
τ ( Y ∩ ∩
i ∈ I
X i ) ≥ min
J ⊆ I τ ©  « Y ∩ ∩
j ∈ J
∂ X j ª ® ¬ .
In particular, τ ( ∩ i ∈ I X i ) ≥ min J ⊆ I τ ( ∩ j ∈ J ∂ X j ) .
The theorem will follow from induction from the following proposition.
Pr oposition 3 §2 2 . [88 ; Theorem 2.4 ] . Let X , Y ⊆ Ò n be closed subsets. Then
τ ( Y ∩ X ) ≥ min { τ ( Y ) , τ ( Y ∩ ∂ X ) } .
Proof of Theorem 3 §2 1 . By Proposition 3 §2 2 , Theorem 3 §2 1 is true for # I = 1 . Assume that
Theorem 3 §2 1 is true for # I ≤ k , we will show that then it is true for # I ≤ k + 1 .
Let I = I 0 ∪ { i 0 } . Then, by Proposition 3 §2 2 ,
τ ( Y ∩ ∩
i ∈ I
X i ) = τ ( ( Y ∩ ∩
i ∈ I 0
X i ) ∩ X i 0 )
≥ min { τ ( Y ∩ ∩
i ∈ I 0
X i ) , τ ( ( Y ∩ ∩
i ∈ I 0
X i ) ∩ ∂ X i 0 ) } .
Since # I 0 ≤ k , by induction hypothesis,
τ ( Y ∩ ∩
i ∈ I 0
X i ) ≥ min
J ⊆ I 0
τ ©  « Y ∩ ∩
j ∈ J
∂ X j ª ® ¬

3 §2 Condition and Homology in Semialgebraic Geometry 111
and, also by induction hypothesis,
τ ( ( Y ∩ ∩
i ∈ I 0
X i ) ∩ ∂ X i 0 ) = τ ( ( Y ∩ ∂ X i 0 ) ∩ ∩
i ∈ I 0
X i )
≥ min
J ⊆ I 0
τ ©  « ( Y ∩ ∂ X i 0 ) ∩ ∩
j ∈ J
∂ X j ª ® ¬ = min
J ⊆ I 0
τ ©  « Y ∩ ∩
j ∈ J ∪ { i 0 }
∂ X j ª ® ¬ .
The first is the minimum over subsets of I that exclude i 0 and the second the minimum over
those that contain i 0 . Hence, taking the minimum of both of them, we prove the theorem for
# I = k + 1 . By the induction principle, the proof is finished. □
Proof of Proposition 3 §2 2 . Let z ∈ ∆ Y ∩ X and x , ˜
x ∈ Y ∩ X be different points such that
dist ( z , x ) = dist ( z , ˜
x ) = dist Y ∩ X ( z ) . As z is arbitrary, it is enough to check that
dist Y ∩ X ( z ) ≥ min { τ ( Y ) , τ ( Y ∩ ∂ X ) }
since, then, taking the infimum over z the desired result follows. We consider three cases:
1) x , ˜
x ∈ ∂ X, 2) x < ∂ X, and 3) ˜
x < ∂ X.
Case 1). In this case, x and ˜
x are also the nearest points to z in Y ∩ ∂ X ⊆ Y ∩ X and so
dist Y ∩ X ( z ) = dist Y ∩ ∂ X ( z ) ≥ τ ( Y ∩ ∂ Y )
which shows our inequality.
Case 2). In this case, x is in the interior of X and so there is some ε > 0 such that
Y ∩ X ∩ B ( x , ε ) = Y ∩ B ( x , ε ) ,
i.e., we cannot distinguish Y ∩ X and Y locally around x . Further, Y ∩ B ( x , ε ) ⊆ B ( x , ε ) \
B ( z , dist X ∩ Y ( z )) , as otherwise x would not be one of the nearest points in Y ∩ X to z , which
contradicts the way x was chosen.
Let
u = z − x
∥ z − x ∥ ,
which is the unit vector of the line joining x to z . By the above paragraph, we must have
π Y ∩ X ( x + t u ) = π Y ( x + t u ) ∈ B ( x , ε ) \ B ( z , dist X ∩ Y ( z ))
for t ∈ [0 , min { ε /2 , τ ( Y ) } ) . Thus π Y ( x + t u ) = x , as this is the nearest point in B ( x , ε ) \
B ( z , dist X ∩ Y ( z )) to x + t u . Therefore τ ( X , x ; u ) > 0 and, by Theorem 3 §1 6 (3),
τ ( Y ) ≤ τ ( Y , x ) ≤ τ ( Y , x ; u ) .
Also τ ( Y , x ; u ) ≤ dist ( x , z ) = dist Y ∩ X ( z ) , by Theorem 3 §1 6 (1), since dist ( z , x ) = dist ( z , ˜
x )
and x , ˜
x ∈ Y. This finishes the proof.
Case 3). This is the same as case 2), but with ˜
x in the place of x .
In all three cases, the given lower bound holds. Hence the proposition is proven. □

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