Re . Real Acad. Cienc. Exac as Fis. Na . Se . A-Ma . (2024) 118:4
h ps://doi.o g/10.1007/s13398-023-01486-8
ORIGINAL PAPER
S a i ied educ ion o singula i ies o gene alized analy ic
unc ions
B. Molina-Sampe 1·J. Palma-Má quez2·F. Sanz Sánchez1
Recei ed: 5 Sep embe 2022 / Accep ed: 14 July 2023
© The Au ho (s) 2023
Abs ac
Gene alized analy ic unc ions a e na u ally de ined in mani olds wi h bounda y and a e buil
om sums o con e gen eal powe se ies wi h non-nega i e eal exponen s. In his pape we
deal wi h he p oblem o educ ion o singula i ies o hese unc ions. Namely, we p o e ha a
ge m o gene alized analy ic unc ion can be ans o med by a ini e sequence o blowing-ups
in o a unc ion which is locally o monomial ype wi h espec o he coo dina es de ining
he bounda y o he mani old whe e i is de ined.
Keywo ds Blowing-up mo phism ·Reduc ion o singula i ies ·Gene alized powe se ies ·
P incipializa ion o ideals
Ma hema ics Subjec Classi ica ion 14E15 ·14P15 ·16W60 ·32C05 ·32S45
Fi s and hi d au ho s a e pa ially suppo ed by he P ojec “Mé odos asin ó icos, algeb aicos y geomé icos
en oliaciones singula es y sis emas dinámicos” (Re .: PID2019-105621GB-100) o he Minis e io de
Ciencia in Spain. Fi s au ho is pa ially suppo ed by he “P og ama de becas pos doc o ales DGAPA” o
he UNAM in Mexico. The second au ho is pa ially suppo ed by Papii Dgapa UNAM IN110520, by he
Is ael Science Funda ion (g an No. 1167/17) and by unding ecei ed om he MINERVA S i ung wi h he
unds om he BMBF o he Fede al Republic o Ge many. This p ojec has ecei ed unding om he
Eu opean Resea ch Council (ERC) unde he Eu opean Union’s Ho izon 2020 esea ch and inno a ion
p og amme (g an ag eemen No. 802107).
BB. Molina-Sampe
bea iz.molina@u a.es
J. Palma-Má quez
[email p o ec ed]
F. Sanz Sánchez
[email p o ec ed]a.es
1Depa amen o de Álgeb a, Análisis Ma emá ico, Geome ía y Topología, Uni e sidad de
Valladolid, Valladolid, Spain
2Weizmann Ins i u e o Science, Reho o , Is ael
0123456789().: V,- ol 123
4 Page 2 o 28 B. Molina-Sampe e al.
1 In oduc ion
In his pape , a gene alized powe se ies (in n a iables and wi h coe icien s in some ing A)
is a powe se ies wi h n- uples o non-nega i e eal numbe s as exponen s and whose suppo
is con ained in a ca esian p oduc o nwell-o de ed subse s o R+={ ≥0}.I iswo h
o men ion ha his condi ion on he suppo is mo e es ic i e (excep o n=1) han he
one used o de ine he Hahn ing A(()),whe eis he g oup Rnwi h he lexicog aphic
o de (and whose elemen s a e also called gene alized powe se ies). In oduced and s udied
by an den D ies and Speissegge in [6], gene alized powe se ies appea in se e al con ex s.
To men ion a ew: as solu ions o di e en ial/ unc ional equa ions; as exp essions o he
Riemann ze a- unc ion (o , mo e gene ally, he Di ichle se ies) in a loga i hmic cha ; as
asymp o ic expansions o Dulac ansi ion maps o ec o ields (see o ins ance [12,13]); in
model heo y and o-minimal geome y ( he pape [6] i sel , o also [17]); as pa ame iza ions
o algeb aic cu es in posi i e cha ac e is ic (see o ins ance [18,p.19]).
Conside ing eal coe icien s, we ha e a na u al no ion o con e gence o gene alized
powe se ies, whose sums p o ide con inuous unc ions on open subse s o he o han Rn
+,
called gene alized analy ic unc ions. They a e he local pieces o build abs ac ( eal) gene -
alized analy ic mani olds, in oduced and de eloped by Ma ín, Rolin and Sanz in [14]. Mo e
p ecisely, a gene alized analy ic mani old is a locally inged space M=(M,GM),whe eM
is a opological mani old wi h bounda y and GMis a shea o con inuous unc ions locally
isomo phic o he shea o gene alized analy ic unc ions on open subse s o Rn
+. Sec ions o
he shea GMa e called hemsel es gene alized analy ic unc ions on M.
The main esul in [14] es ablishes he local educ ion o singula i ies o gene alized ana-
ly ic unc ions, in he spi i o Za iski’s local uni o miza ion heo em o algeb aic a ie ies
[19] o Hi onaka’s e sion o analy ic a ie ies [10]. The s a emen , o mula ed in analo-
gous e ms o hose used in Bie s one–Milman’s pape [4] o eal analy ic unc ions, is he
ollowing:
Local Monomializa ion Theo em [14]. Le be a gene alized analy ic unc ion on Mand
le p∈M. Then he e exis s a neighbou hoodU0o pin M, ini ely many sequences o local
blowing-ups {πi:Mi→U0}
i=1and compac se s Li⊂Misa is ying ha iπi(Li)is a
neighbou hood o pandsuch ha , o e e yi, he o al ans o m i= ◦πiis o monomial
ype a e e y q∈Li(i.e., o some coo dina es x=(x1,x2,...,xn)cen e ed a q,weha e
i=xαU(x)whe e U(0)= 0).
The cen e s o blowing-ups in each sequence πiha e no mal c ossings wi h he bounda y,
bu hey a e de ined only in some open se s o he co esponding mani old. In he s anda d eal
analy ic case, we ha e s onge global monomializa ion esul s ( ypically called Reduc ion
o Singula i ies,see[1,5,7]). They consis , essen ially, in ha in he abo e s a emen , we
can ake jus a single sequence ( =1) and he cen e s o blowing-ups a e globally de ined
closed analy ic submani olds, ha ing no mal c ossings wi h he bounda y.
Such a global esul is no known so a o gene alized analy ic unc ions. The e a e
wo main di icul ies ela ed o he e y no ion o a blowing-up mo phism in he ca ego y
o gene alized analy ic mani olds. On he one hand, a blowing-up depends on he local
coo dina es ha we use o de ine i . Mo e in insically, a blowing-up is no uniquely de ined
and depends on he choice o a s anda diza ion o he mani old (o a leas o an open
neighbou hood o he cen e o blowing-up). Roughly, a s anda diza ion is a subshea OMo
GMsuch ha (M,OM)is a eal analy ic s anda d mani old and om which he shea GMcan
be eco e ed by a na u al comple ion adding gene alized se ies (see [14], we ecall his no ion
below). Secondly, al hough e e y gene alized analy ic mani old is locally s anda dizable,
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S a i ied educ ion o singula i ies... Page 3 o 28 4
he e may exis closed submani olds which do no admi s anda dizable neighbou hoods;
i.e., such submani olds canno be “geome ic” cen e s o a blowing-up (c . [14,Example
3.20]).
Mo ally, a p ocedu e o educ ion o singula i ies o gene alized analy ic unc ions would
need o gua an ee ha , in he p ocess, all closed cen e s suscep ible o be blown-up ha e
s anda dizable neighbou hoods. I his is al eady p o ed and Yis such a cen e , one needs o
show u he mo e ha , among he di e en s anda diza ions a ound Y, he e exis s o which
he co esponding blowing-up π:
M→M educes he “complexi y” o he unc ion.
In his pape , we o e come hese di icul ies o ob ain an in e media e s ep owa ds a global
esul , he so-called s a i ied educ ion o singula i ies. Le us explain i . Fi s , we ecall ha ,
byi s e yde ini ion, hebounda y ∂Mo a gene alizedanaly ic mani oldis ano malc ossing
di iso ; i.e., ∂Mis locally gi en by a ini e union o coo dina e hype planes. Mo eo e , he
numbe o such hype planes a each poin p o ides a na u al s a i ica ion o Mby (s anda d)
analy ic mani olds. A gene alized analy ic unc ion :M→Ris said o be o s a i ied
monomial ype i o any gi en p∈M,i Sis he s a um whe e pbelongs, he e exis s a
local cha (x=(x1,x2,...,xe), y)cen e ed a psa is ying S={x1=x2= ···= xe=0}
and o which
(x,y)=xαU(x,y), whe e α∈Re
+and U(0,y)≡ 0.
Thus, equi ing a unc ion o be o s a i ied monomial ype means o equi e ha i is o
monomial ype only wi h espec o he gene alized coo dina es de e mining equa ions o
he componen s o he bounda y. In pa icula , he condi ion is emp y i p/∈∂M. Also, i is
au oma ic i Shas codimension e=1, aking in he abo e de ini ion α o be he minimum
o he suppo o he se ies de ining wi h espec o he single a iable x=x1.
Ou main esul may be s a ed now as ollows.
Theo em 1.1 (S a i ied Reduc ion o Singula i ies) Le M=(M,GM)be a gene alized
analy ic mani old and le :M→Rbe a gene alized analy ic unc ion. Le p ∈M and
assume ha he ge m o a p is no iden ically ze o. Then, he e exis a neighbou hood Vp
o p in M and a sequence o blowing-ups
(M ,GM )π −1
→(M −1,GM −1)π −2
→ ··· π1
→(M1,GM1)π0
→(Vp,GM|Vp)
such ha he pull-back := ◦π0◦···◦π −1∈GM (M )is o s a i ied monomial ype.
Mo eo e , he cen e o each blowing-up πj, wi h j =0,1,..., −1, can be chosen o be
he closu e o a codimension wo s a um in Mj,whe eM
0:= Vp.
Ou p oo o Theo em 1.1 is cons uc i e in he sense ha each cen e , as well as he s an-
da diza ion used o de ine he espec i e blowing-up a each s ep, can be gi en explici ly in
e ms o he exp ession o in some ini ial coo dina es o Ma p. Mo eo e , each blowing-
up mo phism is locally exp essed as a pu ely monomial map be ween wo domains o Rn
+
in sui able cha s. Consequen ly, all he p ocess o s a i ied educ ion o singula i ies can
be desc ibed using only combina o ics om he s a ing da a gi en simply by he minimal
suppo (see Sec .2below) o a gene alized powe se ies ep esen ing a p. The da um
o minimal suppo is closely ela ed o ha o he New on polyhed on o a unc ion in he
s anda d analy ic case and he e o e, ou esul should be compa ed wi h he combina o ial
educ ion o singula i ies s a ed in Molina’s pape [15]. Al hough i has been a sou ce o
inspi a ion o us, we canno apply di ec ly he esul s in [15], mos ly because he e is no
good no ion o “mul iplici y” in he gene alized non-s anda d si ua ion (any powe unc ion
wi h posi i e eal exponen in a gene alized a iable is a genuine change o a iables).
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4 Page 4 o 28 B. Molina-Sampe e al.
We wan o obse e ha Theo em 1.1 is al eady p o ed o dim M=3 in Palma’s pape
[16], bu wi h a di e en s a egy o he choice o he sequence o blowing-ups ( o ins ance,
he cen e s o blowing-ups may be ei he co ne poin s o closu es o one-dimensional s a a).
The pape is s uc u ed as ollows.
In Sec . 2we summa ize he basic no ions and p ope ies o gene alized powe se ies and
o he ca ego y o gene alized analy ic mani olds, using he men ioned e e ences [6]and
[14]. We emphasize he no ion o s anda diza ion, which is c ucial o de ine blowing-ups.
In Sec . 3we in oduce he ca ego y o monomial (gene alized o s anda d) analy ic mani-
olds. The objec s o his subca ego y a e mani olds ha ing a leas one co ne and equipped
wi h an a las o local cha s cen e ed a each co ne poin o which he change o coo dina es
is exp essed as a monomial map be ween domains o he local model Rn
+. We ep esen hese
changes o coo dina es by means o a amily o ma ices o exponen s ( o a simila ea men
see o ins ance [2,3,16]), a combina o ial da a which codi ies uniquely he s uc u al shea
o he mani old. We de ine also he class o monomial mo phisms and he class o monomial
s anda diza ions o monomial mani olds. A e a blowing-up using such a s anda diza ion
wi h a cen e which is he closu e o a s a um (a so-called combina o ial cen e ), we ob ain
again a monomial mani old and he blowing-up mo phism is a monomial mo phism. The
main esul in his sec ion is he abundance o monomial s anda diza ions (P oposi ion 3.16
below). Fu he mo e, we can choose such a monomial s anda diza ion wi h a p esc ibed local
exp ession a a gi en co ne poin . Mo ally, local s a egies o educ ion o singula i ies a e
suscep ible o be “globalized”. We end his sec ion by in oducing a special class o monomial
mani olds, hose ob ained om a gi en one by a sequence o blowing-ups wi h combina o ial
cen e s, and using only monomial s anda diza ions. Such a sequence is called a monomial
s a and he amily o such s a s is called he monomial “ oû e é oilée”, a e minology ha
e okes he one in oduced by Hi onaka in [10,11] o sequences o local blowing-ups in
complex analy ic geome y.
In Sec . 4we p o ide a p oo o he main Theo em 1.1. Fi s ly, we p o e a esul abou
p incipaliza ion o ini ely gene a ed monomial ideal shea es in a gi en monomial mani old.
This esul (see Theo em 4.5 below) can be seen as a e sion o ou ca ego y o a well known
esul on p incipaliza ion o ideals in he algeb aic o s anda d analy ic si ua ion (see o
ins ance Gowa d’s pape [9] o a simple p oo , o see also Fe nández-Duque’s pape [8] o
a simila s a emen conce ning he esonances elimina ion o singula i ies o codimension-
one analy ic olia ions). Taking in o accoun ha i su ices o ob ain he p incipaliza ion only
a he co ne poin s, such a esul can also be ega ded as a globaliza ion o he algo i hm
desc ibed in an den D ies and Speissegge ’s pape (see [6, Lemma 4.10]) ha educes he
numbe o elemen s in he minimal suppo o a gene alized powe se ies by monomial
ans o ma ions o he a iables.
Al hough we use ce ain elemen s and a gumen s o ha esul , and despi e o wha we
ha e said abo e conce ning he possibili y o globalize a “local s a egy”, ou p oo he e
equi es a di e en con ol in a ian .
Once we ha e he p incipaliza ion o monomial ideal shea es, he main heo em is con-
cluded easily in he case we s a wi h a co ne poin p∈M. In his case, he sequence
π0◦π1◦···π −1 o Theo em 1.1 is ac ually a s a in he oû e é oilée o e he ge m o M
a p. Finally, he gene al case p∈∂Mis educed o he case o a co ne poin , using ha
a ound p he e is a p oduc s uc u e o a neighbou hood o a co ne poin imes a s anda d
analy ic mani old wi hou bounda y.
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S a i ied educ ion o singula i ies... Page 5 o 28 4
2 P elimina ies
Wesumma izehe e hebasicno ionsabou heca ego yo gene alizedanaly ic mani oldsand
blowing-up mo phisms in i , in oduced by Ma ín, Rolin and Sanz in [14]. These mani olds
a e buil om con e gen gene alized powe se ies, ex ensi ely s udied in a pape by an den
D ies and Speissegge [6].
2.1 Fo mal and con e gen gene alized powe se ies
Deno e by R+=[0,∞). Tuples o a iables a e deno ed by X,Y,Z, e c., and we implici ly
assume ha uples wi h di e en name ha e no common a iables. I Xhas ncomponen s,
we say ha Xis an n- uple and so on.
Le X=(X1,X2,...,Xn)be an n- uple o a iables and le Abe an in eg al domain. A
o mal gene alized powe se ies wi h coe icien s in A in he a iables Xisamaps:Rn
+→A,
w i en as
s=
λ∈Rn
+
sλXλ,whe e Xλ=Xλ1
1Xλ2
2···Xλn
n o λ=(λ1,λ
2,...,λ
n)
and sλ:= s(λ) ∈A, such ha i s suppo Supp(s):= {λ∈Rn
+:sλ= 0}is con ained in
a ca esian p oduc o nwell-o de ed subse s o R. The se o all such o mal gene alized
powe se ies, deno ed by A[[X∗]], wi h he usual addi ion and p oduc ope a ions o powe
se ies has an s uc u e o an A-algeb a which is also an in eg al domain. Mo eo e , i Ais
a ield, hen A[[X∗]] is a local algeb a (see [6, Co olla y 5.6]), wi h maximal ideal gi en by
m={s∈A[[X∗]] : s0=0}. No e ha A[[X∗]] is no noe he ian, in ac , he ideal mis
ne e ini ely gene a ed.
The minimal suppo o a powe se ies s∈A[[X∗]] is he subse Suppmin(s)⊂Supp(s)
composed o he minimal uples o Rn
+wi h espec o he (pa ial) di ision o de ≤d, ha is
(λ1,λ
2,...,λ
n)≤d(μ1,μ
2,...,μ
n)i and only i λi≤μi, o alli∈{1,2,...,n}.
The condi ion imposed on he suppo o a powe se ies sallows o show ha he mini-
mal suppo Suppmin(s)is ini e (see [6, Lemma 4.2]). As a consequence, sadmi s a ini e
monomial p esen a ion:
s=
λ∈Suppmin(s)
XλUλ(X),
whe e Uλ∈A[[X∗]] sa is ies Uλ(0)= 0, o any λ∈Suppmin(s). Deno e by m(s)=
#Suppmin(s). When m(s)=1 o , equi alen ly, he monomial ep esen a ion o shas a single
e m, we say ha sis o monomial ype.
In his pape , we a e in e es ed in eal gene alized powe se ies, ha is A=R, bu we use
di e en ings when we wan o dis inguish some a iables and pu he o he s in o he coe i-
cien s. To be p ecise, i Yand Za e uples o kand n−k a iables, espec i ely, we conside
R[[(Y,Z)∗]] as a p ope R-subalgeb a o R[[Y∗]][[Z∗]] by he na u al monomo phism
s=
(λ,μ)∈Rn
+
aλμYλZμ→ sZ=
μ∈Rn−k
+
AμZμ,whe e Aμ=
λ∈Rk
+
aλμYλ.(1)
I p :Rn→Rn−kdeno es he na u al p ojec ion on o he las n−kcoo dina es, o any
powe se ies s∈R[[(Y,Z)∗]] we ha e he inclusion Suppmin(sZ)⊂p (Suppmin(s)),andas
123

4 Page 6 o 28 B. Molina-Sampe e al.
a consequence we ge he inequali y
m(sZ)≤m(s). (2)
Le us w i e R[[Y,Z∗]] o deno e he subalgeb a o R[[(Y,Z)∗]] composed by he so-
called eal mixed powe se ies: hose o mal eal gene alized powe se ies sin he a iables
(Y,Z), such ha he inclusion Supp(s)⊂Nk×Rn−k
+holds, o equi alen ly, such ha
sZ∈R[[Y]][[Z∗]].
Gi en an n- uple o a iables Xand a poly adius ρ=(ρ1,ρ
2,...,ρ
n)∈Rn
>0,deno eby
R{X∗}ρ he subalgeb a o R[[X∗]] consis ing on hose powe se ies s o which
sρ:= 
λ∈Supp(s)
|sλ|ρλ<∞.
The union o he R{X∗}ρalong all he possible poly adius ρ∈Rn
>0is again a subalgeb a
R{X∗}⊂R[[X∗]], and i s elemen s a e called ( eal) con e gen gene alized powe se ies.
We ha e ha R{X∗}is also a local algeb a, whose maximal ideal is gi en by m∩R{X∗}.
I Y,Za e uples o kand n−k a iables, espec i ely, and ρ∈Rn
>0is a poly adius, an
elemen s∈R[[Y,Z∗]] ∩ R{(Y,Z)∗}ρgi es ise o a con inuous unc ion
s:Pρ
k,n−k→R
x=(x1,x2,...,xn)→ λsλxλ,(3)
whe e Pρ
k,n−k=(−ρ1,ρ
1)×(−ρ2,ρ
2)×···×(−ρk,ρ
k)×[0,ρ
k+1)×···×[0,ρ
n)⊂
Rk×Rn−k
+,called hesum o he powe se ies s. Mo eo e , sis ealanaly ica anypoin in he
in e io o Pρ
k,n−kand i s ge m a 0∈Rnis uniquely de e mined by he se ies s.Wede ine he
con e gen mixed powe se ies o be he elemen s o R{Y,Z∗}:=R[[Y,Z∗]] ∩ R{(Y,Z)∗}.
2.2 S anda d and gene alized analy ic mani olds
Le Vbe an open subse o Rn
+and le g:V→Rbe a con inuous unc ion. Gi en a poin
p=(p1,p2,...,pn)∈V, conside Ip:= {i:pi=0}⊂{1,2,...,n}, and pu =#Ip
and k=n−.Wesay ha gis gene alized analy ic (o jus G-analy ic)a pi he e exis s
s∈R{Y,Z∗},whe eYis a k- uple and Zis an - uple, such ha o any x=(x1,x2,...,xn)
in a su icien ly small neighbou hood o 0in Rk×R
+,weha e
g(p1+xσ(1),p2+xσ(2),...,pn+xσ(n))= s(x1,x2,...,xn),
whe e σis a pe mu a ion o he se {1,2,...,n}sa is ying he ela ion j∈Ipi and only i
σ(j)∈{k+1,k+2,...,n}. We say ha gis gene alized analy ic in V i i so a e e y poin
pin V. In he de ini ion abo e, he se ies sis uniquely de e mined by he ge m o ga p,
up o pe mu a ion o he a iables Yand Z, sepa a ely. Thus, he se o ge ms o gene alized
analy ic unc ions a pde ines an R-algeb a isomo phic o R{Y,Z∗}. On he o he hand, i g
is a gene alized analy ic unc ion a some poin p∈Rn
+, hen i is so in a neighbou hood o p
in Rn
+. Summa izing, he assignmen Gn:V→ Gn(V),whe eVis an open subse o Rn
+and
Gn(V)is he se o gene alized analy ic unc ions in V, is a shea o R-algeb as o con inuous
unc ions o e Rn
+, whe e he s alks Gn,pa e local algeb as. Mo eo e Gncon ains he shea
Ono analy ic unc ions, whe e On(V)is he R-algeb a o eal unc ions in Vwhich ex end
o eal analy ic unc ions on some open neighbou hood o Vin Rn.
Wi h his o malism, and aking as local models he locally inged spaces On:= (Rn
+,On)
and Gn:= (Rn
+,Gn), we de ine bo h he ca ego ies o s anda d and gene alized ( eal)
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S a i ied educ ion o singula i ies... Page 7 o 28 4
analy ic mani olds (wi h bounda y and co ne s). The objec s in hese ca ego ies a e called
O-mani olds and G-mani olds, espec i ely. In o de o ea bo h oge he we w i e A o make
e e ence ei he o Oo o G,andA o e e ei he o Oo G.AnA-mani old o dimension nis
a locally inged space M=(M,AM),whe eMis a second coun able Hausdo opological
space ( he unde lying space)andAMis a subshea o he shea C0
Mo ge ms o con inuous
eal unc ions on M( he s uc u al shea ), which is locally isomo phic o he local model
An. Tha is, gi en p∈M he e is an open neighbou hood Vo pin M, an open subse U
o Rn
+and a homeomo phism ϕ:V→Uinducing an isomo phism o he locally inged
spaces
(ϕ, ϕ#):(V,AM|V)∼
−→ (U,An|U),
whe e ϕ#
p:An,ϕ(p)→AM,pis gi en by he composi ion g→ g◦ϕ(as ge ms). A
mo phism be ween wo A-mani olds is jus a mo phism as locally inged spaces, induced by
composi ion wi h con inuous maps on he unde lying spaces (wi h an abuse o language, we
equen ly iden i y mo phisms wi h he co esponding con inuous maps). A couple (V,ϕ)in
he abo e condi ions is called a local cha o Ma p, he componen s x=(x1,x2,...,xn)
o he isomo phism ϕ:V→Ua e local coo dina es a p, and a amily o local cha s
{(Vj,ϕj)}j∈Jsuch ha M=∪
j∈JVjis an a las o M.
Le M=(M,AM)be an A-mani old. No e ha he unde lying space Mis a opological
mani old wi h bounda y, deno ed by ∂M, and ha he es ic ion (M ∂M,AM|M ∂M)is a
s anda d analy ic mani old wi hou bounda y (consequen ly, gene alized analy ic mani olds
wi hou bounda y a e also s anda d). Also he e is a na u al s a i ica ion SMo Mdesc ibed
as ollows. I p∈M,and(V,ϕ)is a local cha a p, he numbe epo anishing coo dina es
in ϕ(p)(equal o #Iϕ(p)) does no depend on he local cha (V,ϕ)chosen (see [14]). In ha
way, he e is a well-de ined map
e:M→{0,1,...,n},p→ ep,
which is uppe semi-con inuous. The elemen s o SMa e he connec ed componen s o he
ibe s o e.Gi enS∈SM, le us w i e eS:= ep,whe epis any poin in S. Obse e ha
(S,AM|S)is a s anda d analy ic mani old o dimension n−eS. In pa icula , he bounda y
∂Mco esponds exac ly wi h he poin s p∈Mwi h ep>0, ha is, ∂Mis equal o he union
o s a a o dimension s ic ly smalle han n.Weha ealso ha ,∂Mis a no mal c ossings
di iso wi h espec o he s uc u al shea . Tha is, o each p∈∂M, he e exis s a local
cha (V,ϕ)o Ma psuch ha
∂M∩V={q∈V:x1(q)·x2(q)···· ·xep(q)=0},
whe e (x1,x2,...,xn)a e he coo dina es associa ed o ϕ.
Example 2.1 Le ¯
Okbe he shea o eal (s anda d) analy ic unc ions in Rk. The locally
inged space (Rk,¯
Ok)is a gene alized and s anda d analy ic mani old, wi h a single cha
ψk:Rk→(0,∞)kde ined by (a1,a2,...,ak)→ (ea1,ea2,··· ,eak).
We obse e a his poin ha he p oduc is de ined in he ca ego y o A-mani olds. Tha
is, gi en wo gene alized o s anda d analy ic mani olds M1=(M1,AM1)and M2=
(M2,AM2)o dimensions nand m, espec i ely, he e is a na u al A-mani old o dimension
n+m, ha we deno e by M1×M2=(M1×M2,AM1×M2), unique up o isomo phism,
sol ing he “p oduc uni e sal p ope y”. Wi hou oo much de ail, he shea AM1×M2is
cons uc ed as ollows. Gi en a poin (p,q)∈M1×M2and wo coo dina e cha s ϕ1:
123
4 Page 8 o 28 B. Molina-Sampe e al.
V1→U1and ϕ2:V2→U2a pand q espec i ely, we ha e ha
AM1×M2,(p,q)={ ◦(ϕ1×ϕ2)(p,q): ∈An+m,(p,q)},
whe e (p,q)=(ϕ1(p), ϕ2(q)).
Example 2.2 The p oduc (Rk,¯
Ok)×(Rn−k
+,An−k),whe eA∈{O,G}, has a na u al s uc-
u e o A-mani old by means o he homeomo phism ψk×id, whe e ψkhas been in oduced
in Example 2.1. We e e o his p oduc by w i ing (Rk×Rn−k
+,Ak,n−k).
Rema k 2.3 Le us conside a poin p∈Mwi h ep=kand le (V,ϕ) be a local cha
o Ma p. Up o pe mu a ion, we can assume ha ϕ(p)=(a1,a2,...,ak,0,...,0)wi h
ai= 0 o alli∈{1,2,...,k}. We can spli he local coo dina es xde ined by ϕin wo
g oups x=(y,z),whe ey=(y1,y2,...,yk)a e s anda d analy ic unc ions a pand
z=(zk+1,zk+2,...,zn)a e gene alized unc ions. By means o ansla ions y
i=yi−ai
in he analy ic coo dina es we ob ain a new isomo phism
ϕ:V→ (ψk×id)−1(ϕ(V)) ⊂Rk×Rn−k
+.
We conside also ϕas a coo dina e cha cen e ed a p in he sense ha ϕ(p)=0∈
Rk×Rn−k
+, and we usually assume ha ou cha s a e cen e ed cha s.
Le us ecallnow heexp essionincoo dina eso hecon inuousmapsinducingmo phisms
o gene alized unc ions (de ails in [14, P oposi ion 3.16]). Conside wo gene alized analy ic
mani oldsM1=(M1,GM1)andM2=(M2,GM2)and acon inuous unc ionφ:M1→M2
inducing a mo phism be ween M1and M2.Gi enp∈M1and q=φ(p)∈M2, ake
(Vp,ϕp),(Wq,ψ
q)cha scen e eda pandq, espec i ely.Followingno a ioninRema k2.3,
deno e by yand z he ks anda d and n−kgene alized coo dina es de ining ϕp, espec i ely.
Up o pe mu a ion, we can assume also ha he i s kcoo dina es de ining ψqa e s anda d
and he o he n−ka e gene alized. Then, he j- h componen ˜
φjo ˜
φ=ψq◦φ◦ϕ−1
pis a
gene alized analy ic unc ion and o j=k+1,k+2,...,n,weha e ha
˜
φj=zλjUj(y,z), Uj(0,0)= 0,λ
j∈Rn−k
+ {0}.(4)
Mo eo e , i φinduces an isomo phism, we ha e ha φis a homeomo phism, n=n,
k=k, hemap ∈Rk→ (˜
φ1( ,0), ˜
φ2( ,0),..., ˜
φk( ,0)) is an analy ic isomo phism, and,
i we w i e λj=(λj,1,λj,2,...,λj,n−k)in Eq. (4), up o a pe mu a ion o coo dina es zwe
ha e
λj,j−k>0,λ
j, =0,∈{1,2,...,n−k} { j−k},(5)
o all j=k+1,k+2,...,n.
We end his sec ion in oducing some no a ion and de ini ions conce ning he s a a o he
na u al s a i ica ion SM.Gi enas a umSin SM,deno ebyS he closu e o Sin M,and
de ine dim(¯
S):= dim(S). We w i e ZM:= {S⊂M:S∈SM}.Fo j=0,1,...,n,
deno e by Zj
M he se o elemen s in ZMwi h dimension j, ha is
Zj
M={¯
S∈ZM:eS=n−j}.
The elemen s o Z0
Ma e he s a a o dimension 0, and a e called co ne poin s, he elemen s
o Z1
Ma e called edges and he elemen s o Zn−1
Ma e called componen s o ∂M.No e ha
∂Mis he union o i s componen s.
Fo each Z∈ZM, we deno e by ZM(Z) he subse o ZMwhose elemen s a e con ained
in Z, and o each j=0,1,...,n, we w i e Zj
M(Z)=ZM(Z)∩Zj
M. We w i e o sho
123
S a i ied educ ion o singula i ies... Page 9 o 28 4
p∈Z0
Mins ead o {p}∈Z0
M, and when no con usion a ises, we will pu Zins ead o ZM,
Zjins ead o Zj
M,e c.
2.3 Monomial complexi y along s a a
We in oduce in his sec ion he concep o monomial complexi y along a s a um and he
de ini ion o s a i ied monomial ype unc ion.
Le us conside a gene alized analy ic mani old M=(M,GM)and a s a um So i s
na u al s a i ica ion S. Take a local cha (V,ϕ)o Mcen e ed a some p∈S, w i e e=eS
and k=dim S=n−e. We can spli he coo dina es de ining ϕ, up o eo de hem, as (y,z),
whe ey=(y1,y2,...,yk)a es anda danaly iccoo dina esin S∩Vandz=(z1,z2,...,ze)
a e gene alized unc ions such ha S∩V={q∈V:z1(q)=z2(q)=··· = ze(q)=0}.
Sh inking Vi necessa y, he cha ϕp o ides an isomo phism
p
ϕ:R{Y,Z∗}→GM,p,s→ s◦ϕ,
whe e Yand Za e kand e uples, espec i ely, and sis he sum o he powe se ies s
in oduced in Eq. (3). Gi en ∈GM,pand s∈R{Y,Z∗} he mixed powe se ies such ha
p
ϕ(s)= , we deno e
SuppS( ;ϕ) =Supp(sZ)⊂Re
+,Suppmin,S( ;ϕ) =Suppmin(sZ)⊂Re
+,(6)
whe e sZ∈R{Y}{Z∗}has been in oduced in Eq. (1).
Lemma 2.4 Le S be a s a um in Swi h e =eS. Take an open subse U o M such ha
U∩S=∅, and a unc ion ∈GM(U). Conside wo local cha s (V1,ϕ
1)and (V2,ϕ
2),
cen e ed a p and q espec i ely, wi h p,q∈S∩U. The e exis s a uple (γ1,γ
2,...,γ
e)∈
Re
>0such ha (λ1,λ
2,...,λ
e)∈Suppmin,S( q;ϕ2)i and only i (γ1λ1,γ
2λ2,...,γ
eλe)∈
Suppmin,S( p;ϕ1).
P oo Using ha Sis pa h connec ed and by compac ness o a gi en pa h om p o q,we
can educe he p oblem o he case whe e bo h poin s pand qbelong o he same connec ed
componen Wo U∩V1∩V2. W i e y=(y1,y2,...,yn−e),z=(z1,z2,...,ze),and
also ¯
y=(¯y1,¯y2,..., ¯yn−e),¯
z=(¯z1,¯z2,...,¯ze), whe e, up o eo de ing, (y,z)a e he
coo dina e unc ions associa ed o ϕ1and (¯
y,¯
z)a e he ones associa ed o ϕ2,insuchaway
ha y|S∩V1,¯
y|S∩V2a e analy ic coo dina es in W∩S. Tha is, we ha e
W∩S={z1=z2=···= ze=0}={¯z1=¯z2=···= ¯ze=0}.
In iew o Eqs. (4)and(5), up o eo de ing he a iables z, he change o coo dina es
ϕ2◦ϕ−1
1sa is ies, o any j=1,2,...,n−eand =1,2,...,e, ha ¯yj=gj(y,z),and
¯z=zγ
h(y,z),whe egj,ha e gene alized analy ic unc ions such ha y→ gj(y,0)is
a s anda d analy ic non-cons an unc ion, γ>0andh(0,0)= 0. We summa ize hese
exp essions by w i ing ¯
y=gand ¯
z=zγh.I 2:= Suppmin,S( q;ϕ2)={μ1,μ
2,...,μ
},
he exp ession o in coo dina es (¯
y,¯
z)is
|W=¯
zμ11(¯
y,¯
z)+¯
zμ22(¯
y,¯
z)+···+¯
zμ  (¯
y,¯
z),
whe e j(¯
y,0)≡ 0, o any j=1,2,..., . Applying he change o coo dina es in o de
o ge he exp ession o in (y,z),weob ain
|W=zγμ
11(y,z)+zγμ
22(y,z)+···+zγμ
 (y,z), k(y,z)=hμkk(g,zγh),
123
4 Page 16 o 28 B. Molina-Sampe e al.
3.3 Abundance o s anda diza ions o monomial mani olds
In his sec ion, we de ine m-s anda diza ions, we gi e a cha ac e iza ion o hei combina-
o ial da a and we p o e a esul o abundance o m-s anda diza ions o a ixed monomial
G-mani old.
Le us ix a monomial gene alized analy ic mani old (M,a).Alocal m-s anda diza ion
o (M,a)a a co ne poin pis jus an m-cha upde ined in he whole open se V
p,such
ha i xp∈a, henup◦x−1
pis gi en by monomial ela ions o he o m
up,i=xαp,i
p,i,whe e αp,i∈R>0, o all i∈Ip.(12)
We ep esen his change o coo dina es by means o he map αp:Ip→R>0de ined
by i→ αp,i. In ha way, he change o coo dina es up◦x−1
pis codi ied by he ma ix o
exponen s Dαp:Ip×Ip→R>0, whe e we ecall ha (once an o de in Ipis ixed) Dαpis
a diagonal ma ix wi h he elemen s αp,iin he diagonal.
De ini ion 3.12 Anm-s anda diza ion o (M,a)is a pai (O,b),whe eOis a s anda diza ion
o Mand b={up}p∈Z0is a monomial a las o N=(M,O)such ha upis a local m-
s anda diza ion o (M,a) o e e y co ne poin p∈Z0.Thecombina o ial da a o an
m-s anda diza ion (O,b)is he collec ion o maps (O,b)={αp}p∈Z0.
Rema k 3.13 I (O,b)and (O,b)a e m-s anda diza ions o (M,a), henbnecessa ily ha
b=bas we ha e al eady no ed in Rema k 3.3. No e also ha he m-s anda diza ion (O,b)
is comple ely de e mined by he combina o ial da a (O,b).
Lemma 3.14 A collec ion o maps ={αp:Ip→R>0}p∈Z0is he combina o ial da a o
an m-s anda diza ion o (M,a)i and only i o any pai o co ne poin s p,q∈Z0 he
ollowing ela ions hold:
αp, =γpq
αq,, o all ∈Ip∩Iq,(13)
whe e γpq is he weigh connexion unc ion om p o q.
P oo Le us assume i s ha =(O,b),whe e(O,b)is an m-s anda diza ion o (M,a).
Le us deno e N=(M,O)and le C(N,b)be he combina o ial da a o he monomial s anda d
analy ic mani old (N,b).In iewo Eq.(11), i is enough o p o e Eq. (13) o wo co ne
poin s pand qconnec ed h ough a compac edge Y. Le us conside he m-cha s up,uq∈b
a pand q, espec i ely. The change o coo dina es up◦u−1
qis codi ied by a ma ix o
exponen s A=Apq
Y∈C(N,b). This change mus be s anda d analy ic in i s domain o
de ini ion uq(V
p∩V
q)=R×Rn−1
+, and his implies
Ai∈Z+,(A−1)j∈Z+, o all i∈Iq,j∈Ip,∈IY.(14)
Le C=Cpq
Y∈C(M,a)and αp,α
q∈(O,b).No e ha Ais ob ained as he p oduc
A=DαqCD−1
αp:Iq×Ip→R.
When ∈Ip∩Iq=IY, in iew o Lemmas 3.7 and 3.9,weha e
A =γpq
αq,
αp,
∈Z+,(A−1) =(Aqp
Y) =γqp
αp,
αq,
=αp,
γpq
αq,
=1/A ∈Z+,
123

S a i ied educ ion o singula i ies... Page 17 o 28 4
which shows A =(A−1) =1. F om he e we ge αp, =γpq
αq,, and hence sa is ies
Eq. (13)aswewan ed.
Assumenow ha sa is iesEq.(13) o anypai o co ne poin s p,q∈Z0.A eachco ne
poin p∈Z0, conside he m-cha upde ined on V
psuch ha he change o coo dina es
up◦x−1
psa is ies up,j=xαp,j
p,j, o all j∈Ip,whe eαp∈and xp∈a.In ha way,we
ge a new monomial a las b={up}p∈Z0o M. Le us see ha he changes o coo dina es
uq◦u−1
pa e s anda d analy ic o any pai o co ne poin s pand q.In iewo Eq.(9)i
is enough o suppose ha pand qa e connec ed h ough an edge Y. De ining he ma ix
A=DαqCD−1
αp, he change o coo dina es uq◦u−1
pis gi en by
uq,i=
j∈Ip
uAij
p,j, o any i∈Iq.
I su ices o show ha Asa is ies he condi ions in Eq. (14). Indeed, i Ai∈Z+ o i∈Iq
and o all ∈IY, henuq,iin he abo e equa ion is s anda d analy ic in e ms o he a iables
upin he domain V
p∩V
q={up,ip= 0}∩{uq,iq= 0}( he same in e changing pand qi
(A−1)j∈Z+ o j∈Iqand any ∈IY). Applying Lemma 3.7 we ge ha A =0and
(A−1) =0, o all ,∈IYwi h = . Mo eo e , he same lemma assu es ha Ciq=0
and ha (Cip)−1=Cqp
ip=0, o all ∈IY; hence, o any such index ∈IYwe ob ain
Aiq=Ciqαq,iq/αp, =0,(A−1)ip=(Cip)−1αp,ip/αq, =0.
Again by Lemma 3.7 we ge
A =Cαq,
αp,
=γpq
αq,
αp,
,(A−1) =(C−1)αp,
αq,
=γqp
αp,
αq,
,
o all ∈IY. Using Lemma 3.9 and Eq. (13) we conclude A =(A−1) =1. As a
conclusion, he a las bde ines a s anda d analy ic s uc u e N=(M,O)o e M,whe e
M=(M,GM); hus O⊂GMis a s anda diza ion o M. Mo eo e , by de ini ion o b,we
ha e ha (O,b)is an m-s anda diza ion o (M,a)wi h (O,b)=.
In he sequel, a collec ion o maps ={αp:Ip→R>0}p∈Z0is called ealizable o (M,a)
i Eq. (13) holds o any pai o co ne poin s p,q∈Z0.
De ini ion 3.15 Le upbe a local m-s anda diza ion o (M,a)a a gi en co ne poin p∈Z0.
An ex ension o upis a (global) m-s anda diza ion (O,b)o (M,a)such ha up∈b;we
sayalso ha (O,b)ex endsup.Wedeno ebyE(up) he se o ex ensions o up.
P oposi ion 3.16 Le (M,a)be a monomial gene alized analy ic mani old. Then:
(a) The e is a bijec ion be ween he se o m-s anda diza ions o (M,a)and RN
>0,whe eN
is he numbe o bounda y componen s o ∂M.
(b) Gi en a co ne poin p ∈Z0and a local m-s anda diza ion upa p, he e is a bijec i e
map RN−n
>0→E(up), whe e n is he dimension o M.
P oo We s a wi h he p oo o he i s asse ion (a). Le Ibe he se o indices labelling
he componen s o ∂M, ha is∂M=i∈IEi,whe eN=#I, and le us ix a collec ion o
co ne poin s q={qi}i∈Iin such a way ha qi∈Ei o each i∈I. Gi en a map β∈RI
>0,
we ake β={αp:Ip→R>0}p∈Z0 o be he amily o maps de ined by
αp, =γpq
β, o all p∈Z0and ∈Ip.
123
4 Page 18 o 28 B. Molina-Sampe e al.
Le us see ha βis a ealizable amily o maps. Fix wo co ne poin s pand q,andle
∈Ip∩Iq.ByEq.(11)weha e ha γpq
γqq
=γpq
. Mo eo e , by he de ini ion o αq,
and as a consequence o Lemma 3.9,weha e ha β=γqq
αq,. Then we ob ain
αp, =γpq
β=γpq
γqq
αq, =γpq
αq,
which is he equi ed condi ion o β o be ealizable. Now, in iew o Lemma 3.14, he e
exis s a unique m-s anda diza ion (Oβ,bβ)wi h (Oβ,bβ)=β.Finally,weshow ha he
map
q:RI
>0→m-s anda diza ions
o (M,a),β→ (Oβ,bβ)
is a bijec ion. Indeed, i β= β,weha e ha β= βand hence (Oβ,bβ)= (Oβ,bβ)
aking in o accoun Rema k 3.13. On he o he hand, gi en an m-s anda diza ion (O,b)wi h
combina o ial da a ={αp}p∈Z0,weha e ha (O,b)=q(β),whe eβis de ined by
βi=αqi,i, o alli∈I. The p oo o (a) is inished.
Le us p o e now hesecond asse ion (b).Deno eby αp:Ip→R>0 hemapo exponen s
de ining up, ha is
up,i=xαp,i
p,i, o all i∈Ip,whe e xp∈a.
Conside he injec i e map iαp:RI Ip
>0→RI
>0,de inedby
δ→ iαp(δ) := βδ,whe e βδ
i=δii i∈I Ip,
αp,ii i∈Ip.
Take a collec ion o co ne poin s qp={qi}i∈Isuch ha qi=p, o eachi∈Ip,andqi∈Ei,
o each i∈I Ip. Using he no a ions in i em a) abo e, we ha e ha qp(β) ∈E(up)i and
only i β|Ip=αp, o equi alen ly β=iαp(β|I Ip). In o he wo ds, we ha e he equali y
E(up)=Im(qp◦iαp),
and hence we ha e he bijec ion RI Ip
>0→E(up)mapping δin o qp(iαp(δ)). We inish jus
by no ing ha #Ip=n.
3.4 The monomial Voû e E oilée
In his sec ion we gi e he de ini ion o m-combina o ial blowing-up and we in oduce he
concep o “monomial oû e é oilée” o e an m-mani old, whose elemen s, called m-s a s,
a e sequences o monomial blowing-ups s a ing om ha m-mani old. The e minology is
inspi ed by Hi onaka [10,11].
Le (M,a)be a monomial gene alized analy ic mani old. An m-combina o ial cen e o
blowing-up o (M,a)is a ipe (Z,O,b),whe eZis a combina o ial geome ic cen e
o Mand (O,b)is an m-s anda diza ion o (M,a). Gi en such an m-combina o ial cen e
(Z,O,b), we conside he blowing-up πξ:Mξ→Mwi h cen e ξ=(Z,O).Le Ibe an
index se labelling he componen s o ∂M. We w i e ∞/∈I o label he excep ional di iso
E∞:= π−1
ξ(Z), and we pu Iξ=I∪{∞}as an index se o he componen s o ∂Mξ.Mo e
p ecisely, gi en i∈I, i ep esen s bo h he bounda y componen Eio ∂Mand i s s ic
ans o m
E
i=π−1
ξ(Ei Z)⊂∂Mξ,
123
S a i ied educ ion o singula i ies... Page 19 o 28 4
belonging o Zn−1
Mand Zn−1
Mξ, espec i ely. The index ∞∈Iξ ep esen s E∞∈Zn−1
Mξ.
P oposi ion 3.17 The e is a monomial a las aξo Mξin such a way ha πξde ines a
mo phism o monomial G-mani olds om (Mξ,aξ) o (M,a).
P oo Take a co ne poin pin Mξand le p=πξ(p). No e ha pis a co ne poin in M.
Le xp∈abe he m-cha o he a las aa p. We dis inguish wo si ua ions:
Casep/∈E∞.Weha e ha Ip=Ipand he blowing-up πξinduces an isomo phism
be ween V∗
pand V∗
p. We ake a ine coo dina es x
po e V
pde ined by
˜x
p,i=xp,i◦πξ|V∗
p, o all i∈Ip.
Thus, he exp ession o πξin coo dina es x
pand xpis pu ely monomial. This exp ession
can be codi ied wi h he ma ix o exponen s Bp:Ip×Ip→R≥0gi en by
Bp(i,j)=δij,i,j∈Ip,(15)
whe e δij is he K onecke del a symbol. In o he wo ds, Bp=D1Ip.
Casep∈E∞.Weha eIp=Ip { j}∪{∞}, o somej∈IZ(see o ins ance [15]
o de ails in he combina o ial ea men o blowing-ups). By hypo hesis, he pai (O,b)is
an m-s anda diza ion o (M,a); in pa icula , bis a monomial a las o he s anda d analy ic
mani old N=(M,O). Using his in o ma ion, oge he wi h he de ini ion o blowing-
up cen e ed a ξ, we ge ha he e exis s an m-cha x
pde ined in V
psuch ha he map
x
p◦πξ◦x−1
pis pu ely monomial wi h associa ed ma ix o exponen s Bp:Ip×Ip→R+
gi en by
( ,s)→ ⎧
⎨
⎩
1i =sand ∈Ip { j},
αp,j/αp, i s=∞ and ∈IZ,
0 o he wise.
(16)
whe e αp∈(O,b). Wi h an app op ia e o de o ows and columns, Bpcan be seen as he
uppe iangula ma ix
⎛
⎝
Idn−s0 0
0Ids−1a
0 0 1
⎞
⎠∈Rn×n
+,
whe e s=#IZand a∈Rs−1
>0is a column ec o whose en ies a e de ined by he quo ien s
αp,j/αp, , wi h ∈IZ { j}.
The collec ion aξ={x
p}p∈Z0
ξwi h Z0
ξ=Z0
Mξis hus a monomial a las in Mξ.Mo e-
o e , he blowing-up πξinduces a mo phism om (Mξ,aξ) o (M,a)and he associa ed
combina o ial da a is Bπξ={Bp}p∈Z0
ξ.
F om now on, gi en an m-combina o ial cen e o blowing-up (Z,O,b) o a monomial
gene alized analy ic mani old (M,a), and he blowing-up mo phism πξ:Mξ→M,
wi h cen e a ξ=(Z,O),wealwaysconside Mξendowed wi h he monomial a las aξ
cons uc ed in P oposi ion 3.17. Mo eo e , we also w i e
πξ:(Mξ,aξ)→(M,a),
oemphasize ha hemo phismπξis conside ed also as a mo phism o monomial gene alized
analy ic mani olds, and we call i an m-combina o ial blowing-up o (M,a). The associa ed
123
4 Page 20 o 28 B. Molina-Sampe e al.
combina o ial da a Bπξ={Bp}p∈Z0
ξo his mo phism has been made explici in Eq. (15),
o poin s p∈Z0
ξwi h p/∈E∞andinEq.(16), o poin s p∈Z0
ξ(E∞).
De ini ion 3.18 Le (M,a)be a monomial gene alized analy ic mani old. An m-s a o e
(M,a)is he composi ion σ=π0◦π1◦···◦π −1o a ini e sequence o m-combina o ial
blowing-ups. Tha is
σ:(M ,a )π −1
−−→ (M −1,a −1)π −2
−−→··· π0
−→ (M0,a0)=(M,a),
whe e o each k=0,1,2,..., −1, he mo phism πkis an m-combina o ial blowing-up
o (Mk,ak). The in ege and he monomial gene alized analy ic mani old (M ,a )a e
called, espec i ely, he age and he end o he m-s a σ. The collec ion Vm
(M,a)o all he
m-s a s o e (M,a)is called he monomial oû e é oilée o (M,a).
4 S a i ied educ ion o singula i ies ia p incipaliza ion o m-ideals
We de o e his sec ion o in oducing he concep o m-ideal, in o de o p o e a heo em o
p incipaliza ion. Wi h his esul we p o e he s a i ied educ ion o singula i ies o a global
unc ion de ined in gene alized analy ic mani olds admi ing a monomial s uc u e. Finally,
we apply his esul o p o e he main esul o his pape s a ed in Theo em 1.1.
4.1 P incipaliza ion o m-ideals
Le us ix a monomial gene alized analy ic mani old (M,a),whe eM=(M,GM).
Take a global gene alized analy ic unc ion ∈GM(M)and wo co ne poin s p,q∈Z0.
Le xq,xq∈abe he m-cha s a pand q, espec i ely, and le Cpq ∈C(M,a)be he ma ix
o exponen s codi ying he change o coo dina es xq◦x−1
p. The ela ion be ween he suppo s
o a pand qwi h espec o hese coo dina es is gi en by:
Suppp( ;xp)={λqCpq :λq∈Suppq( ;xq)}⊂RIp
+.(17)
De ini ion 4.1 A gene alized analy ic global unc ion m∈GM(M)is said o be an m- unc ion
in (M,a)i o each p∈Z0, he eisamapλp:Ip→R+, such ha
m|V
p=xλp
p,whe e xp∈a.
The combina o ial da a o mis he lis Lm={λp}p∈Z0.
Le us conside an m- unc ion min (M,a)wi h combina o ial da a Lm={λp}p∈Z0.By
Eq. (17), o any pai o co ne poin s p,q∈Z0we ha e he ela ion λp=λqCpq,whe e
Cpq ∈C(M,a). In pa icula , we ge ha
λp, =λq,γpq
, o any ∈Ip∩Iq,(18)
whe e γpq is he weigh ed connexion unc ion om p o q. Indeed, in iew o Eq. (11), i is
enough o check Eq. (18) o he case whe e pand qa e connec ed h ough an edge Y.Fo
his case, i holds as a consequence o Lemma 3.7.
Rema k 4.2 Gi en a lis o maps L={λp:Ip→R+}p∈Z0sa is ying λp=λqCpq, o any
pai o co ne poin s p,q∈Z0, he e exis s an m- unc ion msuch ha Lm=L.
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S a i ied educ ion o singula i ies... Page 21 o 28 4
De ini ion 4.3 A ini ely gene a ed m-ideal in (M,a)is a shea o ideals J⊂GMgene a ed
by ini ely many m- unc ions. Tha is,
J=m1GM+m2GM+···+mkGM=: (m1,m2,...,mk),
whe e m1,m2,...,mka e m- unc ions called m-gene a o s o J.
No a ion 4.4 Le Ibe a ini e index se and le Abe a ini e subse o RI. We deno e by Amin
he se o elemen s in A ha a e minimal wi h espec o he di ision o de ≤din RI.
Le Jbe an m-ideal in (M,a)wi h se o m-gene a o s G={m1,m2,...,mk}. Fo each
i=1,2,...,k, le us w i e Lmi={λi
p}p∈Z0. Gi en a co ne poin p∈Z0and xp∈a
he m-cha a p, he es ic ion J|V
pis an m-ideal in he m-co ne (M|V
p,xp)wi h se o
m-gene a o s equal o G|V
p:= {m1|V
p,m2|V
p,...,mk|V
p}. Conside he se
G,p:= {λ1
p,λ
2
p,...,λ
k
p}⊂RIp
+.
No e ha i (G,p)min ={μ1
p,μ
2
p,...,μ
kp
p}, hen
J|V
p=(xμ1
p
p,xμ2
p
p,··· ,xμkp
p
p). (19)
The shea o ideals Jis called locally p incipal i a each poin a∈M, hes alkJa⊂GM,a
is a p incipal ideal. Using he de ini ion o m-ideal, i is enough o ask his p ope y o he
co ne poin s. In e ms o he se in oduced abo e, we ha e ha Jis locally p incipal i and
only i (G,p)min is a single on o any p∈Z0.
Le mbe an m- unc ion in (M,a)and ake an m-s a σ:(M,a)→(M,a).The o al
ans o m σ∗m=m◦σis a again an m- unc ion in (M,a).Mo ep ecisely,i p∈Z0
M
and p=σ(p), henλ
p∈Lσ∗mis gi en by
λ
p=λpBσ
p,(20)
whe e λp∈Lmand Bσ
p∈Bσis he ma ix o exponen s codi ying σa p.I Jis an
m-ideal gene a ed by G={m1,m2,...,mk}, hen he o al ans o m σ∗Jis also an m-ideal
in (M,a)gene a ed by σ∗G:= {σ∗m1,σ∗m2,...,σ∗mk}.
The main esul in his sec ion is he ollowing one abou p incipaliza ion o m-ideals.
Theo em 4.5 Le Jbe a ini ely gene a ed m-ideal in a monomial gene alized analy ic man-
i old (M,a). The e exis s an m-s a σ∈Vm
(M,a)such ha σ∗Jis locally p incipal.
To p o e his heo em, we can educe ou sel es o he case whe e Jis gene a ed by wo
m- unc ions by conside ing a clea ini e ecu ence and he ollowing lemma.
Lemma 4.6 Le J=(m1,m2,...,mk)be an m-ideal in (M,a). Assume ha J s :=
(m ,ms)is locally p incipal o any pai o indices ,s∈{1,2,...,k}.ThenJis locally
p incipal.
P oo Assume ha he e is a poin p∈Z0such ha Jpis no p incipal. The e exis indices
,s∈{1,2,...,k}such ha λ
p∈Lm and λs
p∈Lmsa e no compa able o he di ision
o de ≤din RIp. No e ha G s,p=(G s,p)min ={λ
p,λ
s
p},whe eG s ={m ,ms},and
hence J s is no locally p incipal, which is a con adic ion. 
123

4 Page 22 o 28 B. Molina-Sampe e al.
Thecaseo wogene a o s.Le us assume ha J=(m,n)is an m-ideal in (M,a)gene a ed
by wo m- unc ions and w i e Lm={λp}p∈Z0and Ln={μp}p∈Z0.
We in oduce i s se e al de ini ions, mainly inspi ed by he “b-in a ian ” in oduced in
[6] by an den D ies and Speissegge .
Le Z∈Zn−2be a codimension wo combina o ial geome ic cen e o M.Weknow
ha he index se IZhas jus wo elemen s, say IZ={i,j}.Le p∈Z0(Z)be a co ne poin
in Z. We say ha Zis uncoupled o Ja p i
(λp,i−μp,i)(λp,j−μp,j)<0.
We say ha Zis uncoupled o Ji i is so a each p∈Z0(Z).
Lemma 4.7 A combina o ial geome ic cen e Z ∈Zn−2is uncoupled o Ji and only i
he e is a co ne poin q ∈Z0(Z)such ha Z is uncoupled o Ja q.
P oo Assume ha Zis uncoupled o Ja a co ne poin q∈Z0(Z)and ake any o he
poin p∈Z0(Z).ByEq.(18), we ha e λp, =λq,γpq
, o all∈Ip∩Iq,whe eγpq is
he weigh ed connexion unc ion om p o q.Sincep,q∈Z, we know ha IZ={i,j}⊂
Ip∩Iq.Then
(λp,i−μp,i)(λp,j−μp,j)=γpq
iγpq
j(λq,i−μq,i)(λq,j−μq,j)<0,
since γpq
i>0andγpq
j>0. Hence Zis uncoupled o Jalso a p, and we conclude ha Z
is uncoupled o J.
Obse e ha i p∈Z0and he e a e no uncoupled cen e s o Jpassing h ough p, hen
we necessa ily ha e ha λp≤dμpo μp≤dλp, ha isJpis a p incipal ideal.
De ini ion 4.8 Le Jbe he amily o codimension wo combina o ial geome ic cen e s in
M ha a e uncoupled o J, and de ine he in a ian o J o be In J:= #J.
We ha e ha In J=0 i and only i Jis locally p incipal. Thus, he objec i e now is o ind
an m-s a σ∈Vm
(M,a)such ha In σ∗J=0.
Suppose ha In J>0and ixZ∈J. Take a co ne poin p∈Zand pick a local
m-s anda diza ion upo (M,a)a pde ined by he map αp:Ip→R>0. We say ha upis
adap ed oJwi h espec oZ i
αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=0.
A global m-s anda diza ion (O,b={up}p∈Z0)o (M,a)is said o be adap ed o Jwi h
espec oZ i upis adpa ed o Jwi h espec o Z o e e y p∈Z0(Z).
Lemma 4.9 An m-s anda diza ion (O,b)is adap ed o Jwi h espec o Z i and only i he e
is a co ne poin q ∈Z such ha uq∈bis adap ed o Jwi h espec o Z.
P oo Deno e =(O,b). Assume ha he e is a co ne poin q∈Zsuch ha uq∈bis
adap ed o Jwi h espec o Z. Take any o he co ne poin p∈Z. In iew o he ealizabili y
o es ablished in Lemma 3.14 we know ha αq, =γqp
αp,, o all∈Ip∩Iq.Since
p,q∈Zwe ha e ha IZ={i,j}⊂Ip∩Iq. Then, by Eq. (18), we ge
αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=γpq
jαq,jγpq
i(λq,i−μq,i)
+γpq
iαq,iγpq
j(λq,j−μq,j)
=γpq
iγpq
j[αq,j(λq,i−μq,i)
+αq,i(λq,j−μq,j)]=0.
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S a i ied educ ion o singula i ies... Page 23 o 28 4
As a consequence, he local m-s anda diza ion up∈bis adap ed o Jwi h espec o Z
a p. We conclude ha (O,b)is adap ed o Jwi h espec o Z.
A codimension wo combina o ial cen e o blowing-up ξ=(Z,O,b)is adap ed o Ji
Z∈Jand (O,b)is an m-s anda diza ion adap ed o Jwi h espec o Z. The nex esul
assu es he exis ence o such a cen e .
Lemma 4.10 Assume ha In J>0. Then, he e exis codimension wo combina o ial cen e s
o blowing-up adap ed o J.
P oo By de ini ion In J>0 i and only i J=∅. Fix an elemen Z∈Jand le us
see ha he e a e m-s anda diza ions adap ed o Jwi h espec o Z. In iew o Lemma 4.9,
i is enough o p o e he exis ence o an m-s anda diza ion (O,b)adap ed o Ja a co ne
poin p∈Z. Fix any co ne poin p∈Z.SinceZis uncoupled o J,wecanassume,up o
exchanging he indices iand j, ha
i=λp,i−μp,i>0,and j=μp,j−λp,j>0.
Take αp:Ip→R>0 o be a map such ha αp,i=iand αp,j=j, and ake he m-cha
up=xαp
pde ined in V
p. Any m-s anda diza ion ex ending upis adap ed o Jwi h espec
o Za he poin pbecause o he de ini ion o αp. Mo eo e , such an ex ension exis s as a
consequence o P oposi ion 3.16.
We conclude by applying ini ely many imes he ollowing esul .
P oposi ion 4.11 Le J=(m,n)be an m-ideal wi h In J>0. Gi en an m-combina o ial
cen e o blowing-up ξ=(Z,O,b)adap ed o J, he blowing-up πξ:Mξ→Mcen e ed
a ξsa is ies In π∗
ξJ=In J−1.
P oo Le us w i e Zξ=ZMξ,π=πξ,E∞=π−1(Z)and IZ={i,j}. Deno e also
Lm={λp}p∈Z0,Ln={μp}p∈Z0,Lπ∗m={λ
p}p∈Z0
ξ,Lπ∗n={μ
p}p∈Z0
ξ.
Le Tbeacodimension wocombina o ialgeome iccen e inMdi e en om Z.Deno e
by ST he s a um in SMsuch ha ST=T. The closu e To π−1(ST)is a codimension wo
geome ic cen e in Mξha ing index se IT=IT={ ,s}. Gi en a co ne poin p∈T,
le p=πξ(p).Weha e
λp, =λp, ,μ
p, =μp, ,λ
p,s=λp,s,μ
p,s=μp,s,
in iew o he ela ion be ween λp,λpand μp,μpes ablishedin Eq. (20), and he exp ession
o Bπ
p∈Bπgi eninEqs.(15)and(16).Then,weha e ha T∈π∗Ji andonly i T∈J,
ha is Tis uncoupled o π∗Ji and only i Tis uncoupled o J. Le us see now ha any
elemen in π∗Jis among he ones conside ed be o e. Tha is, le us show ha he e is no
codimension wo combina o ial geome ic cen e Zuncoupled o π∗Jcon ained in E∞.
Take a codimension wo combina o ial geome ic cen e Z⊂E∞and a poin p∈
Z0
ξ(Z). In iew o Lemma 4.7, i is enough o p o e ha Zis no uncoupled o π∗Ja p.
Mo e p ecisely, i we w i e IZ={k,∞},wewan oshow ha
(λp,k−μp,k)(λp,∞−μp,∞)≥0.
Le us conside p=π(p)and he local da a αp∈(O,b). The co ne poin pbelongs
o Z, and he a ine coo dina es up∈bde ine a local m-s anda diza ion adap ed o Jwi h
123
4 Page 24 o 28 B. Molina-Sampe e al.
espec o Za p, ha is, we ha e he ela ion αp,j(λp,i−μp,i)+αp,i(λp,j−μp,j)=0.
We know ha Ip=Ip { j}∪{∞}, up o exchanging he indices iand j. Hence, he ma ix
B:= Bp:Ip×Ip→R+sa is ies, using Eq. (16):
B
 =1, o ∈Ip { j},B
j∞=1,B
i∞=αp,j
αp,i
=μp,j−λp,j
λp,i−μp,i
,B
s =0 o he wise.
By Eq. (20) we ge he ela ions λp,k=λp,k,μp,k=μp,k,and
λp,∞=λp,j+B
i,∞λp,i=λp,iμp,j−λp,jμp,i
λp,i−μp,i
=μp,j+B
i,∞μp,i=μp,∞.
Thus (λp,k−μp,k)(λp,∞−μp,∞)=0, and we a e done. 
4.2 S a i ied educ ion o singula i ies in monomial mani olds
We use he esul o p incipaliza ion o m-ideals in o de o p o e he ollowing s a emen :
P oposi ion 4.12 Le (M,a)be a monomial gene alized analy ic mani old wi h M=
(M,GM). Gi en a gene alized analy ic unc ion ∈GM(M), he e is an m-s a σ∈Vm
(M,a)
such ha he pull-back = ◦σis o s a i ied monomial ype.
Fo he p oo o P oposi ion 4.12 we associa e o a ini ely gene a ed m-ideal J ,and
we p o e ha he p incipaliza ion o J gi es ise o he s a i ied educ ion o singula i ies
o .
Gi en q∈Z0and λq∈Suppq( ;xq), i makes sense o de ine he m- unc ion mλqas he
one ha ing he collec ion o maps Lmλq={λqCpq}p∈Z0as a combina o ial da a, by Rema k
4.2 and Eq. (17). The m-ideal J associa ed o is he ideal shea gene a ed by he ini e se
o m- unc ions
G =
q∈Z0mλq:λq∈Suppmin,q( ;xq).
By de ini ion, no ice ha o any co ne poin q∈Z0,weha e
(G ,q)min =Suppmin,q( ;xq). (21)
whe e he he no a ion G ,qwas in oduced in Sec . 4.1 abo e.
Lemma 4.13 Gi en an m-s a τ:(M,a)→(M,a), we ha e τ∗J =J ,whe e =
◦τ.
P oo In iewo Eq. (19),i isenough o p o e ha o anyco ne poin p∈Z0
M heequali y
(τ∗G ,p)min =(G ,p)min holds. Deno e o sho 1=τ∗G ,pand 2=G ,p.
Fix a poin p∈Z0
Mand conside p=τ(p). W i e := Suppmin,p( ;xp),whe exp∈
a,andle Bτ
p∈Bτbe he ma ix o exponen scodi ying τa p.Le := {λpBτ
p;λp∈}.
We p o e ha bo h min
1and min
2a e equal o min.
S ep 1: min
2=min.Recall by Eq. (21) ha min
2=:= Suppmin,p( ;x
p),whe e
x
p∈a.Le ={λ1,λ
2,...,λ
k}⊂RIp, ha is, he unc ion a ound he co ne poin
phas he ini e p esen a ion |V
p=xλ1
pU1+xλ2
pU2+···xλk
pUk,whe eUi(p)= 0, o all
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S a i ied educ ion o singula i ies... Page 25 o 28 4
i=1,2,...,k. Taking in o accoun ha τ(V
p)⊂V
p, he unc ion = ◦τis w i en in
he cha xp∈aas
|V
p=x˜
λ1
p(U1◦τ|V
p)+x˜
λ2
p(U2◦τ|V
p)+···x˜
λk
p(Uk◦τ|V
p),
whe e ˜
λi=λiBτ
p.SinceBτ
pis an in e ible ma ix, we can assu e ha ˜
λ = ˜
λs o any
pai o di e en indices ,s∈{1,2,...,k}. This implies ha ={
˜
λ1,˜
λ2,...,˜
λk}min by
de ini ion o minimal suppo o a p. We a e done, since ={
˜
λ1,˜
λ2,...,˜
λk}.
S ep 2: min
1=min.Recall ha ⊂G ,pand deno e ˜
=G ,p .ByEq.(21)
we know ha o any μ∈˜
 he e exis s λ∈such ha λ≤dμ. No e ha 1=∪˜
,
whe e
˜
:= {μBτ
p;μ∈˜
}.
The e o e, we need only o p o e he ollowing claim: I λ,μ :Ip→R+sa is y λ≤dμ,
hen λBτ
p≤dμBτ
p. Fo ha , i is enough o conside he case whe e τ=πξis a single
m-combina o ial blowing-up wi h cen e ξ=(Z,O,b). Deno e, as usual, E∞=π−1
ξ(Z).
I p/∈E∞o equi alen ly p/∈Z,weha e ha Ip=Ipand Bτ
p=D1Ip, and we a e done.
Assume ha p∈E∞,andle jbe he index in IZsuch ha Ip {∞} = Ip { j}. Deno e
λ=λBτ
pand μ=μBτ
p.ByEq.(16), we ha e ha λ
=λ,μ
=μ, and hus λ
≤μ
,
o all ∈Ip {∞}; whe eas
λ
∞=
∈IZ
αp,j
αp,
λ≤
∈IZ
αp,j
αp,
μ=μ
∞,
whe e αp∈(O,b),aswewan ed. 
P oo o P opos ion 4.12 In iew o Theo em 4.5, we can ake an m-s a σ:(M,a)→
(M,a)such ha σ∗J islocallyp incipal.ByLemma4.13weknowalso ha σ∗J =J ◦σ.
Hence, since G ◦σis a se o gene a o s o J ◦σ,weha e ha (G ◦σ,p)min is a single on
o all p∈Z0
M. Finally, by Eq. (21) we ob ain
mp( )=#Suppmin,p( ;x
p)=#(G ◦σ,p)min =1
o all p∈Z0
M.Sinceais a monomial a las, we know ha M=p∈Z0
MV
p. Thus,
gi en a s a um S∈SM, he e is a co ne poin p∈Z0
Msuch ha p∈¯
S.In iewo he
ho izon al s abili y p ope y o he monomial complexi y es ablished in Lemma 2.6,wege
mS( )≤mp( )=1. We conclude ha is o s a i ied monomial ype. 
4.3 P oo o he main s a emen
We end end he e he p oo o he s a i ied educ ion o singula i ies o gene alized analy ic
unc ions, as s a ed in Theo em 1.1.
Recall ha we ha e a gene alized analy ic mani old M=(M,GM), and a gene alized
analy ic unc ion ∈GM(M)in M. Gi en a poin p∈M, we wan o p o e ha he e exis
an open neighbou hood V⊂Mo pand a ini e sequence o blowing-ups
σ:M
π −1
−→ M −1
π −2
−→··· π0
−→ M0=(V,GM|V),
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