Publicacions Ma em`a iques, Vol 43 (1999), 127–162.
STRICTLY ANALYTIC FUNCTIONS
ON p-ADIC ANALYTIC OPEN SETS
Kamal Boussa
Abs ac
Le Kbe an algeb aically closed comple e ul ame ic field.
M. K asne and P. Robba defined heo ies o analy ic unc ions in
K, bu when Kis no sphe ically comple e bo h heo ies ha e he
disad an age o con aining unc ions ha may no be expanded
in Taylo se ies in some disks. On o he hand, affinoid heo ies
a e only defined in a small class o se s (union o affinoid se s) [2],
[13] and [17]. He e, we suppose he field K opologically sepa able
(example Cp). Then, we gi e a new defini ion o s ic ly analy ic
unc ions o e a la ge class o domains called analoid se s. Ou
heo y uses he no ion o T-sequence which ca ac e izes analy ic
se s in he sense o Robba. The eby we ob ain analy ic unc ions
sa is ying he p ope y o analy ic con inua ion and which, how-
e e , will admi expansion in powe se ies ( esp. Lau en se ies)
in any disk ( esp. in any annulus). Mo eo e , he algeb a o ana-
ly ic unc ions will be s able by de i a ion. The p ocess consis s
o defining a la ge class o analy ic se s D, and a class o admissi-
ble se s making a co e ing o such a D, so ha we ob ain a shea
on D. We finally gi e an example o diffe en ial equa ion whose
solu ions a e s ic ly analy ic unc ions in an analoid se . Such an
example migh no be in ol ed in heo ies based on affinoid se s.
I. P elimina ies
Le Kbe an algeb aically closed field comple e o an ul ame ic
absolu e alue. We ecall some s anda d no a ions and defini ions.
Gi en a∈Kand >0, d(a, ) ( esp. d(a, −), esp. C(a, )) de-
no es he ci cum e enced disk {x∈K||x−a|≤ }( esp. he non-
ci cum e enced disk {x∈K||x−a|< }, esp. he ci cle {x∈
K||x−a|= }). We call a class o d(a, ) any non-ci cum e enced
disk d(b, −) wi h |a−b|≤ .
Le >0 and >
,Γ(a, , ) ( esp. ∆(a, , )) deno es he
annulus {x∈K| <|x−a|<
}( esp. {x∈K| ≤|x−a|≤ }).
128 K. Boussa
Le Dbe an infini e subse o K. Then R(D) deno es he se o a ional
unc ions h∈K(x) wi h no poles in D. This is a K-subalgeb a o he
algeb a KDo all unc ions om Din o K. Then R(D) is p o ided
wi h he opology UDo uni o m con e gence on D, and is a opological
g oup o his opology. H(D) deno es he comple ion o R(D) o his
opology and i s elemen s a e named he analy ic elemen s on D[4], [6],
[11].
Defini ion. A se Dis said o be analy ic i o e e y ∈H(D) o
e e y a∈Dand >0, he p ope y (x) = 0 whene e x∈d(a, )∩D
implies (x) = 0 whene e x∈D.
In Theo em 0 we ecall he cha ac e iza ion o analy ic se s by using
T-sequences.
Example and ema ks. Le (an)n∈Nbe a sequence in Ksuch ha
|an|<|an+1|∀n∈N, lim
n→∞ |an|= 1 and
∞
n=0
−log |an|<+∞. Le
ρ∈]0,1[ and ( n)n∈Nbe a sequence in ]0,ρ[. Then by [6, P oposi-
ion 36.5], he open se D=d(0,2) (
n∈N
d(an,
n−)) is analy ic.
No ice ha Robba’s defini ion o analy ic unc ions may ex end o
open se s like D. Howe e , in a no sphe ically comple e field such as
Cp, his defini ion gi es unc ions ha may no be expanded in Taylo
se ies in some disks.
On he o he hand, affinoid se s ( esp. “connec ed” affinoid se s i.e.:
in aconnec ed affinoid se s) a e defined by F esnel and Van De Pu in
[7] and we e used o cons uc a heo y o analy ic unc ions defined
by a shea o analy ic elemen s on affinoid se s. Bu , o example, i
we conside he analy ic se Ddefined abo e, we see ha such a se
can’ be co e ed by an inc easing sequence o in aconnec ed affinoid
se s. The e o e, by F esnel and Van De Pu p ocess, one ob ains, o
example, he cha ac e is ic unc ion o he se d(0,2) d(0,1−)asan
analy ic unc ion on D([7, 1.7]). Hence, we see ha he amily o affinoid
se s is oo small o gi e a gene al heo y o analy ic unc ions, sa is ying
he p inciple o analy ic con inua ion.
The aim he e is o cons uc a la ge amily o analy ic unc ions, ich
in p ope ies, and defined on a la ge class o open analy ic se s named
analoid se s, as a e Robba’s se s, bu a oiding he incon enience o con-
aining unc ions ha may no be expanded in Taylo se ies in some
disks in a non-sphe ically comple e field. Pa icula ly, we will see ha
S ic ly analy ic unc ions on p-adic analy ic open se s 129
s ic ly analy ic unc ions sa is y he p inciple o analy ic con inua ion
on any analoid se (Theo em 28).
Nex , we ecall some p elimina ies defini ions and pa icula ly his o
T-sequences.
A sequence (an)n∈Nin Kis said o be an inc easing dis ances sequence
( esp. a dec easing dis ances sequence) i he sequence |an+1 −an|is
s ic ly inc easing ( esp. dec easing) and has a limi l∈R∗+.
A sequence (an)n∈Nis said o be a mono onous dis ances sequence i
i is ei he an inc easing dis ances sequence o a dec easing dis ances
sequence.
A sequence (an)n∈Nin Kis said o be an equal dis ances sequence i
|an−am|=|am−aq|whene e n,m,q∈Nsuch ha n=m=q.
A se Din Kis said o be in aconnec ed i o e e y a∈D, he
mapping Ia om D o R+defined by Ia(x)=|x−a|has an image whose
closu e in R+is an in e al. (In o he wo ds, a se Dis no in aconnec ed
i and only i he e exis aand b∈Dand an annulus Γ(a, 1,
2) wi h
0<
1<
2<|a−b|such ha Γ(a, 1,
2)∩D=∅.)
As usual, gi en a se Ain Kand a poin a∈K, we deno e by δ(a, A)
he dis ance om a o A.
Le Dbe an infini e se in K, and le a∈D.I Dis bounded o
diame e , we deno e by
D he disk d(a, ), and i Dis no bounded,
we pu
D=K. Then,
D Dis known o admi a pa i ion o he o m
(d(ai, −
i)i∈J), wi h i=δ(ai,D) o each i∈J. The disks d(ai, −
i)i∈J,
a e named he holes o D.
1. Mono onous fil e s.
Le a∈
Dand S∈R∗
+be such ha Γ(a, , S)∩D=∅whene e
∈]0,S[ ( esp. Γ(a, S, )∩D=∅whene e >S). We call an inc easing
( esp. a dec easing)fil e o cen e a and diame e S,onD he fil e Fon
D ha admi s o base he amily o se s Γ(a, , S)∩D( esp. Γ(a, S, )∩
D). Fo e e y sequence ( n)n∈Nsuch ha n<
n+1 ( esp. n>
n+1) and lim
n→∞ n=S, i is seen ha he sequence Γ(a, n,S)∩D
( esp. Γ(a, S, n)∩D) is a base o Fand such a base is called a canonical
base [6].
Gi en an inc easing ( esp. a dec easing) fil e Fon Do cen e aand
diame e , we will deno e by PD(F) he se {x∈D||x−a|≥ }
( esp. he se {x∈D||x−a|≤ }). Fu he PD(F) will be named he
D-beach o F.
130 K. Boussa
We call a mono onous fil e on Da fil e which is ei he an inc easing
fil e o a dec easing fil e .
Gi en a mono onous fil e Fwe will deno e by diam(F) i s diame e .
The field Kis said o be sphe ically comple e i each nes ed sequence
o disks has a nonemp y in e sec ion. The field Cp, o example, is no
sphe ically comple e. Howe e , e e y algeb aically closed comple e ul-
ame ic field admi s a sphe ically comple e algeb aically closed ex en-
sion [6].
Le Fbe an inc easing ( esp. a dec easing) fil e o cen e aand di-
ame e Son D. The fil e Fis said o be pie ced i o e e y ∈]0,S[,
( esp. >S), Γ(a, , S)( esp. Γ(a, S, )) con ains some hole Tmo D.
2. Mono onous dis ances holes sequences.
Le a∈
D. Le (Tm,i)1≤i≤s(m)
m∈N
be a sequence o holes o Dwhich
sa isfies δ(a, Tm,i)=dm(1 ≤i≤s(m),m∈N), dm<d
m+1 ( esp. dm>
dm+1), and lim
m→∞ dm=R>0.
The sequence (Tm,i)1≤i≤s(m)
m∈N
is called an inc easing ( esp. dec easing)
dis ances holes sequence ha uns he inc easing ( esp. dec easing) fil-
e Fo cen e a, o diame e R. The fil e Fwill be named he inc eas-
ing ( esp. dec easing) fil e associa ed o he sequence (Tm,i)1≤i≤s(m)
m∈N
.
The D-beach o Fwill be also named he D-beach o (Tm,i)1≤i≤s(m)
m∈N
.
Finally, an inc easing ( esp. dec easing) dis ances holes sequence
will be called a mono onous dis ances holes sequence and he
sequence (dm)m∈Nis called he mono ony o he mono onous dis ances
holes sequence.
Le (Tm,i)1≤i≤s(m)
m∈N
be a mono onous dis ances holes sequence and
o e e y (m, i)(i∈{1,... ,s(m)},m ∈N), le ρm,i = diam(Tm,i).
The numbe lim in
m→∞ ( min
1≤i≤s(m)(ρm,i)) ( esp. lim sup
m→∞
( max
1≤i≤s(m)(ρm,i))) will
be called in e io limi -pie cing ( esp. supe io limi -pie cing) o he se-
quence (Tm,i)1≤i≤s(m)
m∈N
.
I a mono onous holes sequence o diame e has an in e io limi -
pie cing ρ>0 and a supe io limi -pie cing ρ< , i will be said co ec ly
pie ced.
A se Dwill be said o be co ec ly pie ced i e e y mono onous dis-
ances holes sequence o Dwi h a no emp y D-beach is co ec ly pie ced.
S ic ly analy ic unc ions on p-adic analy ic open se s 131
A se Dis said o be well pie ced i δ(D,K D)>0, i.e. he se o
diame e s o holes o Dhas a s ic ly posi i e lowe bound.
3. Weigh ed sequences.
We call a weigh ed sequence a sequence (Tm,i,q
m,i)1≤i≤s(m)
m∈N
wi h
(Tm,i)1≤i≤s(m)
m∈N
a mono onous dis ances holes sequence and (qm,i)1≤i≤s(m)
m∈N
a sequence o nonnega i e in ege s. Gi en m∈Nand i∈{1,... ,s(m)},
qm,i is called he weigh o Tm,i. The D-beach o (Tm,i)1≤i≤s(m)
m∈N
is also
named he D-beach o (Tm,i,q
m,i)1≤i≤s(m)
m∈N
.
The mono onous fil e associa ed o (Tm,i)1≤i≤s(m)
m∈N
, is also called fil e
associa ed o (Tm,i,q
m,i)1≤i≤s(m)
m∈N
.
A weigh ed sequence is said o be co ec ly pie ced i i s associa ed
mono onous dis ances holes sequence is.
Fo e e y m∈N,wepu
Ωm= max
1≤i≤s(m)
qm,i log dm
ρm,i +
j=i
1≤j≤s(m)
qm,j(log dm−log |am,j −am,i|)
.
The sequence (Ωm)m∈Nwill be called pe u ba ions sequence o he
weigh ed sequence.
We will say ha he weigh ed sequence has a pe u ba ions sequence
bounded by λ∈R+, i sup
m∈N
Ωm≤λ. Mo e gene ally we say ha he
weigh ed sequence has a bounded pe u ba ions sequence, i sup
m∈N
Ωm<+∞.
A weigh ed sequence (Tm,i,q
m,i)1≤i≤s(m)
m∈N
will be said o be idempo en
i qm,i = 0 o 1 o all (m, i), (1 ≤i≤s(m),m∈N) and qm,i = 0 o
infini ely many (m, i).
Le S1and S2be wo mono onous dis ances holes ( esp. weigh ed)
sequence. We will say ha S1and S2a e cofil ing i hey a e associa ed
o he same mono onous fil e o K.
We see ha wo cofil ing sequences ha e he same cen e s and he
same D-beach.
132 K. Boussa
Le S=(T
m,i)1≤i≤k(m)
m∈N
and S =(T
m,i)1≤i≤k (m)
m∈N
be wo cofil ing
mono onous dis ances holes sequences, o cen e a. The holes o he se
D=K (∪1≤i≤k(m)
m∈N
T
m,i)∪(∪1≤i≤k (m)
m∈N
T
m,i)
o m a mono onous dis ances sequence which is cofil ing o Sand S.
We will deno e i by S∪S. Thus, S∪S is in he o m (Tm,i)1≤i≤k(m)
m∈N
,
and we ema k ha a hole o S∪S
is ei he a hole o So a hole o
S.
4. T-sequences.
Le S=(Tm,i,q
m,i)1≤i≤s(m)
m∈N
be an inc easing ( esp. dec easing)
weigh ed sequence and o all m∈N, le qm=
s(m)
i=1
qm,i. The weigh ed
sequence Swill be said o be a T-sequence i i sa isfies:
lim
m→∞
sup
1≤j≤s(m)
dm
ρm,j qm,j
i=j
1≤j≤s(m)
dm
|am,i −am,j|qm,i
m−1
n=1 dn
dmqn
=0
( esp.
lim
m→∞
sup
1≤j≤s(m)
dm
ρm,j qm,j
i=j
1≤j≤s(m)
dm
|am,i −am,j|qm,i
m−1
n=1 dm
dnqn
=0).
Rema k. A weigh ed sequence (Tm,i,q
m,i)1≤i≤s(m)
m∈N
is a T-sequence
i and only i
lim
m→∞
−sup
1≤j≤s(m)
qm,j(log dm−log ρm,j)
+
i=j
1≤i≤s(m)
qm,i(log dm−log |am,i −am,j|)
+
m−1
n=1
qn|log dm−log dn|
=+∞.
S ic ly analy ic unc ions on p-adic analy ic open se s 133
Theo em 0. Le Dbe a se in K. Then Dis analy ic i and only i
any T-sequence o Dhas an emp y D-beach.
P oo : By [6, Theo em 38.8], we know ha Dis analy ic i and only
i Dis in aconnec ed and any T-sequence o holes o Dhas an emp y
D-beach. Thus we only ha e o show ha i Dis no in aconnec ed,
hen Dadmi s a T-sequence wi h a no emp y D-beach. Le us suppose
ha Dis no in aconnec ed. Then he e exis a,b∈Dand 1, 2∈R+
such ha 0 <
1<
2<|a−b|and Γ(a, 1,
2)∩D=∅. Hence, we
see ha e e y elemen o Γ(a, 1,
2) belongs o a hole o D. So, i is
easly seen ha Γ(a, 1,
2) admi s a pa i ion Pby a amily o holes o
D.By[12, P oposi ions 1.2 and 2.5] he e exis ∈]0,
2[, u∈d(a, 2)
and an inc easing idempo en T-sequence Swi h holes in P, o cen e u
and diame e . Then since |b−u|≥ , we see ha Sis a T-sequence o
holes o Dwi h a no emp y D-beach.
II. S ic ly analy ic unc ions
1. Pola and T-pola sequences.
Le Dbe in aconnec ed, le a∈K, le >0 and le ρ∈]0, [.
We call an inc easing ( esp. dec easing) pola sequence, o cen e a,o
diame e and sepa a ion ρe e y sequence o he o m (bm,i)1≤i≤k(m)
m∈N
sa is ying: bm,i ∈
D D,∀(m, i)(m∈N,i ∈{1,... ,k(m)}), wi h
|bm,i −a|=|bm,j −a|=dmwhene e i,j∈{1,... ,k(m)},|bm,i −a|<
|bm+1,j −a|( esp. |bm,i −a|>|bm+1,j −a|) whene e 1 ≤i≤k(m)
and 1 ≤j≤k(m+ 1), lim
m→∞ dm= and in
(m,i)=(n,j)|bm,i −bn,j|=ρ. The
sequence (dm)m∈Nis called he mono ony o he pola sequence. We call
he D-beach o (bm,i)1≤i≤k(m)
m∈N
he se D∩(K d(a, −)) ( esp. D∩d(a, )).
A pola sequence (bm,i)1≤i≤k(m)
m∈N
is called T-pola sequence i o some
σ∈]0,ρ] he e exis s a amily (qm,i)1≤i≤k(m)
m∈N
o nonnega i e in ege s such
ha (d(bm,i,σ−),q
m,i)1≤i≤k(m)
m∈N
is a T-sequence.
Rema k. An elemen o a pola sequence is ei he an elemen o D D
o an elemen o a hole o D.
134 K. Boussa
Example. Suppose K=Cp. Le Γ = {x∈K;xpn= 1, o some
n≥0}and le D=K Γ. I is well known ha elemen s o Γ o m a
sequence (bm,i)1≤i≤pm−1(p−1)
m∈N
whe e bm,i lies in he ci cle o cen e 1 and
diame e p−1
pm(p−1) . Then by [5, P oposi ion II.4], (bm,i)1≤i≤pm−1(p−1)
m∈N
is a T-pola sequence.
No ice ha o all <p
−1
p(p−1) he se D ={x∈K;|x−γ|≥ ∀γ∈
Γ}is no analy ic ([5]).
2. Defini ion o an analoid se .
We build up ou heo y o s ic ly analy ic unc ions on he so-called
analoid se s.
Defini ion. Dwill be said o be an analoid i Dsa isfies:
1) E e y T-pola sequence admi s an emp y D-beach.
2) E e y mono onous dis ances holes sequences wi h a no emp y D-
beach has a supe io limi -pie cing s ic ly in e io o i s diame e .
3) Dis open.
Rema k. An analoid se is analy ic. Indeed, suppose ha an analoid
se Dadmi s a T-sequence S=(Tm,i,q
m,i)1≤i≤s(m)
m∈N
wi h a no emp y
D-beach. Le Rbe he diame e o S. Fo e e y (m, i)(m∈N,
i∈{1,... ,s(m)}), le Tm,i =d(am,i,ρ
−
m,i). Since Dis analoid, Shas
a supe io limi -pie cing ρ<R. Hence, wi hou loss o gene ali y we
assume ha diame e o holes o Sa e uppe bounded by σ∈]ρ, R[.
Le (d(bm,j,σ−))1≤j≤l(m)
m∈N
be he sequence o disks o diame e σsuch
ha e e y hole Tm,i (m∈N,1≤i≤s(m)) is included in some
d(bm,j,σ−) and ha e e y disk d(bm,j ,σ−)(m∈N,1≤j≤l(m))
con ains some Tm,i. Fo e e y (m, j)(m∈N,1≤j≤l(m)) we de-
no e by Im,j he se o (m, i) such ha Tm,i ⊂d(bm,j,σ−) and we pu
pm,j =
(m,i)∈Im,j
qm,i. Then, by [6, P oposi ion 35.4], he weigh ed se-
quence S=(d(bm,j,σ−),p
m,j)1≤j≤l(m)
m∈N
is a T-sequence and he e o e D
admi s a T-pola sequence wi h a no emp y D-beach, a con adic ion
wi h he hypo hesis “Dis analoid”.
S ic ly analy ic unc ions on p-adic analy ic open se s 135
3. Examples.
Defini ions. Le Dbe a se ha con ains a leas wo poin s.
Dis called quasi-connec ed ([11]) i o any wo poin s x,y∈D, he
se {|z−x|;z∈K D, |z−x|≤|y−x|} is fini e.
A quasi-connec ed se Dis called egula ([10]) ( esp. comple ely egu-
la [13]) i o any wo poin s x,y∈Dand any ∈|K∗|wi h ≤|y−x|,
he se (K D)∩d(a, ) can be co e ed by coun ably ( esp. fini ely) many
open balls wi h adius .
We ecall ha he no ion o egula quasi-connec ed se was fi s gi en
by K asne ([11]) when he esidue class field o Kis no coun able.
Quasi-connec ed se s (and pa icula y egula and comple ely egula
quasi-connec ed se s) a e analoids. Indeed, on one hand, i is well known
ha quasi-connec ed se s a e open. On he o he hand, since o all x,y
in a quasi-connec ed se D, he se {|z−x|;z∈K D, |z−x|≤|y−x|}
is fini e, we see ha Dhas nei he mono onous dis ances holes sequences
no pola sequence wi h a no emp y D-beach and consequen ly i is an
analoid.
Bu gene ally, analoid se s a e no quasi-connec ed. Indeed, an analoid
may ha e mono onous dis ances holes sequence and pola s sequence wi h
a no emp y beach, which is no ue o quasi-connec ed se s.
Le (an)n∈Nbe an inc easing dis ances sequence in he disk d(0,1) o
limi 1 sa is ying
∞
n=0
−log |an|<+∞and le (ρn)n∈Nbe a sequence in
]0,1[ such ha lim sup
n→∞
ρn<1. Le D=d(0,2) (
n∈N
d(an,ρ
−
n)). Clea ly
he holes o Da e o he o m d(an,ρ
−
n). Acco ding o [6, P oposi-
ion 36.5], such a se is analy ic and hen we see ha Dis an analoid.
Rema ks. i) One can’ ha e an inc easing co e ing o such a se D
by admissible se s ei he in he sense o [F esnel Van de Pu and Mo i a]
o in he sense o [Ka lowski and Ull ich]. Consequen ly, one can’ define
analy ic unc ions on Din hese diffe en senses.
ii) We no ice ha he class o analoid se s is no s able by in e sec ion,
bu we define a subclass o “special” analoid se s which is so and which
makes co e ing o e e y analoid se .
142 K. Boussa
Theo em 8. The p eshea His a shea o he G- opology on D.
P oo : This is a consequence o he ollowing lemma.
Lemma 9. Le U1and U2be co ec ly pie ced closed analy ic se s
such ha U1∩U2=∅.Le 1∈H(U1), 2∈H(U2)such ha
1/U1∩U2= 2/U1∩U2.
Then, he e exis s ∈H(U1∪U2)such ha /Uj= j(j=1,2).
P oo : By Lemma 5 a hole o U1∩U2is ei he a hole o U1o a hole
o U2. Hence Lemma 9 holds by [14, Theo em 8.3].
Rema k. In a nex pape we will s udy p ope ies o his shea .
Le ’s now p o e ha an analoid admi s a co e ing by an inc easing
sequence o D-admissible se s.
Lemma 10. Le S=(d(am,i,ρ
−
m,i),q
m,i)1≤i≤s(m)
m∈N
, and
S=(d(bm,i,ρ
−
m,i),q
m,i)1≤i≤s(m)
m∈N
be wo weigh ed idempo en sequences
o diame e Rand le δ>0and <Rsuch ha ∀m,n∈N,∀i∈
{1,... ,s(m)},∀j∈{1,... ,s(n)}i=j, we ha e δ≤ρm,i ≤ ,δ≤
ρ
m,i ≤ and |am,i −an,j|=|bm,i −bn,j|.
Then Sis a T-sequence i and only i Sis a T-sequence.
P oo : Le m∈Nand le j∈{1,... ,s(m)}. We pu
Am,j =
i=j
1≤i≤s(m)
dm
|am,i −am,j|qm,i
=
i=j
1≤i≤s(m)
dm
|bm,i −bm,j|qm,i
and i (dn)n∈Nis inc easing ( esp. dec easing) we pu
Bm=
m−1
n=1 dn
dmqn
( esp.
Bm=
m−1
n=1 dm
dnqn
).
S ic ly analy ic unc ions on p-adic analy ic open se s 143
By defini ion, S( esp. S)isaT-sequence i and only i
lim
m→∞ sup
1≤j≤s(m)dm
ρm,j qm,j
Am,jBm=0(1)
( esp.
lim
m→∞ sup
1≤j≤s(m)dm
ρ
m,j qm,j
Am,jBm=0.(2)
Acco ding o he hypo hesis o he lemma, ob iously he e exis
α,β∈R+and N∈Nsuch ha
αqm,i ≤dm
ρm,i qm,i
≤βqm,i ,and(3)
αqm,i ≤dm
ρ
m,i qm,i
≤βqm,i ,∀i∈{1,... ,s(m)}.(4)
We now suppose ha Sis a T-sequence; consequen ly Ssa isfies (1).
Bu since each qm,i lies in {0,1}(m∈N,1≤i≤s(m)), by (1) and (3),
o e e y sequence (xm)m∈Nsuch ha xm∈{1,... ,s(m)},weha e
(5) lim
m→∞ Am,xmBm=0.
In pa icula , i (xm)m∈Nis such ha
dm
ρ
m,xmqm,xm
Am,xm= sup
1≤j≤s(m)dm
ρ
m,j qm,j
Am,j,
we see ha using (4) and (5) we ha e
lim
m→∞ sup
1≤j≤s(m)dm
ρ
m,j qm,j
Am,jBm=0.
We ha e a symme ic p oo when Sis a T-sequence.
Defini ion. Two cofil ing weigh ed sequences
S=(d(am,i,ρ
−
m,i),q
m,i)1≤i≤s(m)
m∈N
and S=(d(am,i,ρ
−
m,i),q
m,i)1≤i≤s(m)
m∈N
will be said o be simila i hey ha e he same in e io and supe io
limi -pie cing.
144 K. Boussa
Co olla y 11. Le Sand Sbe simila weigh ed sequences. Then S
is co ec ly pie ced i and only i Sis co ec ly pie ced.
Besides, i Sis co ec ly pie ced, hen Sis an idempo en T-sequence
i and only i Sis an idempo en T-sequence.
Lemma 12. Le U1,U2be in aconnec ed se s such ha
δ(U1,U
2)≤min(diam(U1),diam(U2)).
Then U1∪U2is in aconnec ed.
P oo : Le a,b∈U1∪U2and le 1, 2∈Rsuch ha 0 <
1<
2<
|a−b|. We jus ha e o check ha (U1∪U2)∩Γ(a, 1,
2)=∅.
I a,b∈U1o a,b∈U2, hen since bo h U1and U2a e in aconnec ed,
we ha e U1∩Γ(a, 1,
2)=∅o U2∩Γ(a, 1,
2)=∅and he e o e
(U1∪U2)∩Γ(a, 1,
2)=∅.
Now we assume ha a∈U1and b∈U2. Fi s suppose, 1<diam(U1).
Then he e exis s a∈U1such ha 1<|a−a|≤diam(U1). Bu since
U1is in aconnec ed and a,a∈U1, we ha e U1∩Γ(a, 1,min( 2,|a−
a|)) =∅. Consequen ly (U1∪U2)∩Γ(a, 1,
2)=∅. Now suppose
1≥diam(U1). We fi s assume U2∩
U1=∅; hen o e e y b∈U2∩
U1,
we ha e Γ(a, 1,
2)=Γ(b,
1,
2). So, since U2is in aconnec ed and
0<
1<
2<|b−b|, we ha e U2∩Γ(b,
1,
2)=∅and he e o e
(U1∪U2)∩Γ(a, 1,
2)=∅.
Finally, we assume U2∩
U1=∅. Then i is clea ha δ(U1,U
2)=
in
x∈U
2
(|x−a|). Bu as 1≥diam(U1) and δ(U1,U
2)≤min(diam(U1),
diam(U2)), he e exis s b∈U2such ha |b−a|<
2. Since U2is
in aconnec ed, we ha e Γ(b,|b−a|,
2)∩U2=∅; bu as Γ(a, |b−a|,
2)=
Γ(b,|b−a|,
2)⊂Γ(a, 1,
2), he e o e (U1∪U2)∩Γ(a, 1,
2)=∅.
We ha e a symme ic p oo when a∈U2and b∈U1. This ends he
p oo o Lemma 12.
Lemma 13. Le U1and U2be in aconnec ed se s such ha U1∪U2
is in aconnec ed. Then
U1⊂
U2o
U2⊂
U1.
P oo : Suppose ha
U1⊂
U2and
U2⊂
U1. By ul ame ici y i is
seen ha δ(
U1,
U2)>max(diam(
U1),diam(
U2)) and δ(
U1,
U2)=|a−b|
o all a∈U1and b∈U2. Le 1, 2∈R+be such ha diam(
U1)<
1<
2<|a−b|; hen we see ha Γ(a, 1,
2)∩(
U1∪
U2)=∅. Bu his
con adic s he hypo hesis ha U1∪U2is in aconnec ed.
S ic ly analy ic unc ions on p-adic analy ic open se s 145
Lemma 14. Le U1,U2be in aconnec ed se s such ha U1∪U2is
in aconnec ed. Then a hole o U1∪U2is ei he a hole o U1o a hole
o U2.
Besides, i a hole o U1∪U2is included in
U1and
U2 hen i is included
in a hole o U1and in a hole o U2.
P oo : Le T=d(a, −) be a hole o U1∪U2. By Lemma 13, we can
assume wi hou loss o gene ali y ha
U1⊂
U2. Fi s we suppose ha
T⊂
U1. Then i is ob ious ha Tis simul aneously included in a hole
o U1and in a hole o U2.
Since = in
x∈U1∪U2
(|x−a|) = min( in
x∈U1
(|x−a|),in
x∈U2
(|x−a|)), i is
seen ha Tis ei he a hole o U1o a hole o U2.
Now i we suppose ha T⊂
U1, hen i is clea ha δ(a,
U1)> .
Bu since δ(a,
U1)=δ(a, U1) = in
x∈U1
(|x−a|), hen ob iously we ha e
= in
x∈U2
(|x−a|) and as T⊂
U2,Tis a hole o U2.
Lemma 15. Le Dbe an analoid and le U1,U2be D-admissible
se s wi hou inc easing T-sequence. Fu he , we assume ha U1∪U2is
co ec ly pie ced.
a) I U1∩U2=∅, hen U1∪U2is D-admissible.
b) I δ(U1,U
2) = diam(U1) = diam(U2), hen U1∪U2is D-admiss-
ible.
P oo : By Lemma 12, i U1∩U2=∅o δ(U1,U
2) = diam(U1)=
diam(U2) hen U1∪U2is in aconnec ed. Besides, since U1∪U2is
bounded, closed and co ec ly pie ced, we only ha e o show ha i
is well pie ced, analy ic, ha e e y ci cled hole o Dincluded in
U1∪U2
is s ic ly included in a hole o U1∪U2and ha i Dis pe iphe ally ci -
cled, hen
U1∪U2
D. The wo las s a emen s a e ob ious, because
on one hand a ci cled hole o Dincluded in
U1∪U2is included in a hole
o U1∪U2. Bu by Lemma 14, a hole o U1∪U2is a hole o U1o a
hole o U2. Then, since bo h U1and U2a e D-admissible se s, we ha e
a s ic inclusion. On he o he hand, i Dis pe iphe ally ci cled, hen
bo h
U1and
U2a e s ic ly included in
D. Hence,
U1∪U2is s ic ly
included in
D.
Since U1and U2a e well pie ced, hen by Lemma 14 so is U1∪U2.
We will show ha U1∪U2is analy ic. Indeed le us suppose ha his
is no ue. So, by Theo em 0, U1∪U2admi s a T-sequence Swi h
a no emp y U1∪U2-beach. Bu since U1∪U2is co ec ly pie ced, by
146 K. Boussa
Lemma 3, we may assume ha Sis an idempo en T-sequence wi h a
bounded pe u ba ions sequence.
Lemmas 14 and 4 show ha he e exis s a T-sequence S1o S2whose
holes a e holes o U1o holes o U2 espec i ely. Fo example le us sup-
pose S1 o be his T-sequence. Ob iously S1is dec easing because U1
does no admi inc easing T-sequences. Since U1is analy ic, by Theo-
em 0 we see ha S1has an emp y U1-beach. Hence he U1∪U2-beach
o Sis included in U2. Le abe an elemen o he U1∪U2-beach o S,
hen a∈U2.
a) We assume U1∩U2=∅and ake b∈U1∩U2. In pa icula we ha e
|a−b|>diam(S). By Lemma 14, we see ha om ce ain ank, he
holes o S1a e included in holes o U2. To such a hole o U2we associa e
he sum o he weigh s o holes o S1 ha i con ains. Hence, we ob ain
a weigh ed sequence o holes o U2which is, by [6, P oposi ion 35.4], a
T-sequence. Thus U2admi s a T-sequence wi h a no emp y U2-beach,
which con adic s he hypo hesis ha U2is analy ic.
b) Now we assume ha
(1) δ(U1,U
2) = diam(U1) = diam(U2).
Since S1is dec easing, i is clea ha
(2) diam(S1)<diam(U1).
As a∈U2, by (2), we ha e δ(U1,U
2)≤diam(S1) which is a con adic ion
wi h (1).
Thus U1∪U2is analy ic and he e o e i is D-admissible.
No a ion. Hence o h, Kis supposed opologically sepa able. I is
well known ha such a field is no sphe ically comple e ([14]) and we
see ha Cpsa isfies such a condi ions.
Defini ion. We will call p epie ced fil e on De e y mono onous fil-
e on Dwi h cen e , less hin han a pola sequence.
Rema k. Le Dbe an in aconnec ed se o Kand le λ<diam(D).
Then, since Kis sepa able, he amily o disks d(a, λ−) included in
D
which con ain elemen s o D Dis coun able. Mo eo e , he amily
o disks included in
Dwhose cen e s a e cen e s o p epie ced fil e s o
diame e λis coun able oo.
S ic ly analy ic unc ions on p-adic analy ic open se s 147
Lemma 16. Le a∈Kand >0. Then a pa i ion o d(a, −)by
non-ci cum e enced disks is a single on o infini e.
P oo : Suppose ha he pa i ion is no educed o a single on. Le
d(a, ρ−) be he elemen o he pa i ion con aining a. Then, we see
ha ρ< . Since Kis algeb aically closed, i s alua ion g oup is dense
in R. Then, le b,c∈Ksuch ha ρ<|b−a|<|c−a|< . Le
d(b, ρ−
b) and d(c, ρ−
c) a e he elemen s o he pa i ion con aining band
c espec i ely. Ob iously we ha e d(b, ρ−
b)∩d(c, ρ−
c)=∅. Hence, he
pa i ion o d(a, −) is infini e.
Lemma 17. Le Dbe an analoid. Le d(a, −)be such ha
d(a, )
D.Le λ∈]0, [and λ/∈|K|.Le (d(an,ρ
n−))n∈Nbe he
amily o holes o Dincluded in d(a, −), o diame e supe io o equal
o λ.Le (d(bn,µ
n−))n∈Nbe he amily o holes o Dincluded in d(a, −),
o diame e s ic ly in e io o λ.Le (cn)n∈Nbe a sequence o elemen s
o Dsuch ha , o all n∈N,cnis cen e o a p epie ced fil e o Do
diame e λ.Le (d(dn,λ
−))n∈Nbe he amily o disks which con ain an
elemen o D D.
I (sn)n∈Nand ( n)n∈Na e sequences in R+o limi s ze o, such ha
d(a, −)=
n∈N
(d(an,ρ
n(1 + sn)−)
∪d(bn,λ(1 + n)−)∪d(cn,λ
−)∪d(dn,λ
−)),
hen he e exis s m∈Nsuch ha
d(a, −)=d(am,ρ
m(1 + sm)−)o
d(a, −)=d(bm,λ(1 + m)−).
P oo : Le T1={d(an,ρ
n(1 + sn)−); n∈N}, le T2={d(bn,λ(1 +
n)−); n∈N}, le T3={d(cn,λ
−); n∈N}and le T4={d(dn,λ
−); n∈
N}. We pu T=T1∪T
2∪T
3∪T
4. We will deno e by R he ela ion
defined on Tby URVi he e exis s W∈T such ha U⊂Wand
V⊂W. This ela ion is ob iously seen o be an equi alence ela ion on
T. Fo e e y U∈T, we pu
U=
V∈U
Vwhe e Uis he equi alence class
o U.
We will show ha o each U∈T, he e exis s V∈Usuch ha
V=
Uand ha he e is only one equi alence class wi h espec o he
ela ion R.
148 K. Boussa
Suppose ha o ce ain U∈T we ha e V
U o all V∈U.
The e o e he e exis s a sequence (Vα(n))n∈Nin U, s ic ly inc easing
wi h espec o he inclusion, whose limi diame e s is equal o diam(
U).
Wi hou loss o gene ali y we may assume ha he sequence (Vα(n))n∈N
ei he is in T1o is in T2o is in T3o is in T4.
Fi s , we assume ha Vα(n)∈T
1,∀n∈Nand we w i e
Vα(n)=d(aα(n),ρ
α(n)(1 + sα(n))−),∀n∈N.
Then, since lim
n→+∞sn= 0, we see ha
(1) lim
n→∞ ρα(n)= diam(
U).
By [6, Theo em 3.1], we may ex ac om (aα(n)
)n∈Na sequence (aβ(n)
)n∈N
which ei he is con e gen o is an equal dis ances sequence o is a
mono onous dis ances sequence.
Since ρn≥λ o all n∈Nand d(an,ρ
−
n)∩d(am,ρ
m)=∅, o all
n=m, clea ly he sequence (aβ(n))n∈Ncan’ be con e gen .
I (aβ(n))n∈Nis an equal dis ances sequence o alue A, hen since
(Vα(n))n∈Nis a s ic ly inc easing sequence, we ob iously ha e
(2) A<diam(
U).
Mo eo e , as he holes o Da e disjoin ed, we ha e
(3) ρβ(n)≤A, ∀n∈N.
Then we see ha (2) and (3) con adic (1).
I (aβ(n))n∈Nis a mono onous dis ances sequence, hen by (1) neces-
sa ily he sequence (d(aβ(n),ρ
−
β(n))n∈Nis an inc easing dis ances holes
sequences, o diame e diam(
U) and o supe io limi -pie cing diam(
U).
Then, since he sequence (d(aβ(n),ρ
−
β(n))n∈Nhas a no emp y D-beach
(because Vβ(n)⊂
U⊂d(a, −)
D,∀n∈N), we see ha his con a-
dic s he hypo hesis “Dis an analoid”.
Second, suppose ha Vα(n)∈T
2,∀n∈N. On one hand, since
(Vα(n))n∈Nis s ic ly inc easing, we see ha diam( ˆ
U)>λ. On he o he
hand, since lim
n→+∞ n= 0, we ha e diam( ˆ
U)=λ, which is impossible.
S ic ly analy ic unc ions on p-adic analy ic open se s 149
Finally, (Vα(n))n∈Ncan’ be a s ic ly inc easing sequence o he in-
clusion in T3( esp. T4) because hei elemen s a e o diame e λ.Thus
his finishes p o ing ha o each U∈T, he e exis s V∈Usuch ha
V=
U. Hence we see ha d(a, −) admi s a pa i ion by a amily o
elemen s o T.
Le us suppose ha his pa i ion is no educed o a single on. Then,
by Lemma 16 his pa i ion is infini e. Le ∈]λ, [. By [12, P oposi-
ion 2.5] we deduce he exis ence o an idempo en inc easing T-sequence
S=(d(um, −
m),1)1≤i≤k(m)
m∈N
o diame e and whose elemen s belong
o T.
We will show ha Sis co ec ly pie ced. Indeed, we no ice ha e e y
elemen o Thas a diame e supe io o λ, and he e o e he in e io
limi -pie cing o Sis no ze o. Mo eo e , by defini ion, e e y hole o T3
and T4is o diame e λ. We also ema k ha , since lim
n→+∞ n= 0, hen
e e y mono onous dis ances sequence in T2has a supe io limi -pie cing
equal o λ. On o he hand, since lim
n→+∞sn= 0 and D∩(K d(a, −)) =∅
e e y inc easing dis ances sequence o holes in T1has a no emp y D-
beach, and he e o e i is co ec ly pie ced (because Dis). Consequen ly,
as >λand as T=T1∪T
2∪T
3∪T
4,Sis co ec ly pie ced oo.
Le bbe a cen e o Sand le m∈N.
•I d(um, −
m)∈T
1, hen he e exis s γ(m)∈Nsuch ha
d(um, −
m)=d(aγ(m),ρ
γ(m)(1 + sγ(m))−) and he e o e we pu
d( m, −
m)=d(aγ(m),ρ
−
γ(m)).
•I d(um, −
m)∈T
2, hen he e exis s γ(m)∈Nsuch ha
d(um, −
m)=d(bγ(m),λ(1 + γ(m))−) and he e o e we pu
d( m, −
m)=d(bγ(m),λ
−).
•I d(um, −
m)∈T
3, hen he e exis s γ(m)∈Nsuch ha
d(um, −
m)=d(cγ(m),λ
−). In his case, by hypo hesis, d(cγ(m),λ
−)
is he disk o cen e s o a p epie ced fil e o diame e λ. Then we
see ha he e exis s m∈
D Dsuch ha
(4) |um− m|<|um−b|.
So, i mbelongs o a hole o Do diame e ρ, we pu d( m,
m
−)=
d( m,ρ
−); else (i.e. m∈D D), we pu d( m,
m
−)=d( m,λ
−).
•I d(um, −
m)∈T
4 hen he e does exis γ(m)∈Nsuch ha
d(um, −
m)=d(dγ(m),λ−) and hen we pu d( m, −
m)=d(dγ(m),λ−).
150 K. Boussa
Hence we ha e ob ained a weigh ed sequence S=(d( m, −
m),1)1≤i≤k(m)
m∈N
which is cofil ing o Sand sa is ying
(5) |um−un|=| m− n|,∀m, n ∈N.
We fi s see ha Sis inc easing and has a no emp y D-beach because
d(a, −)
D. Bu since Dis co ec ly pie ced and 0 <λ< ,Sis also
co ec ly pie ced.
Now we see ha bo h Sand Sa e cofil ing, co ec ly pie ced, sa is y
(5) and ha Sis an idempo en T-sequence. Then by Lemma 10, S
is an idempo en T-sequence. The e o e, by Lemma 3, we can assume
ha Shas a bounded pe u ba ions sequence. As each hole o Sei he
is a hole o Do con ains an elemen o D D, hen Lemma 4 shows
ha Dadmi s ei he a T-sequence wi h a no emp y D-beach o a T-
pola sequence wi h a no emp y D-beach. Bu his con adic s he
hypo hesis ha Dhas no T-pola sequences wi h a no emp y D-beach.
So, he pa i ion o d(a, −) is educed o a single on, and he e o e,
he e exis s T∈T such ha T=d(a, −). Bu since >λ, we see ha
he e exis s m∈Nsuch ha ei he d(a, −)=d(am,ρ
m(1 + sm)−)o
d(a, −)=d(bm,λ(1 + n)−). This ends he p oo o Lemma 17.
Lemma 18. Le Dbe an analoid, le a,b∈Dand le >0such ha
≤|a−b|. Then he e exis s a D-admissible Ua, con aining a, o di-
ame e , wi hou inc easing T-sequences and such ha e e y inc easing
dis ances sequence o holes o Ua, is co ec ly pie ced.
P oo : Le λ∈]0,min( , δ(a, K D))[ be such ha λ/∈|K|. Le
(Fn)n∈Nbe he sequence o p epie ced fil e s o diame e λ, secan wi h
d(a, −).
Le (d(an,ρ
n−))n∈Nbe he amily o holes o Dincluded in d(a, −), o
diame e supe io o equal o λ, le (d(bn,µ
n−))n∈Nbe he amily o holes
o D, included in d(a, −), o diame e s ic ly in e io o λ, le (cn)n∈N
be such ha o all n∈N,cnis cen e o Fnand le (d(dn,λ
−))n∈Nbe he
amily o disks o diame e λwhich con ain elemen s o (D D)∩d(a, −).
Fo all n∈N, we pu un=|a−an|.I d(an,ρ
n−) is ci cled, we ha e
un>ρ
n. So, we may choose εn∈]0,1
n+1 [ sa is ying ρn(1 + εn)<u
n.I
d(an,ρ
n−) is no ci cled, we pu εn= 0.
S ic ly analy ic unc ions on p-adic analy ic open se s 151
Fo all n∈N, we pu T1n=d(an,ρ
n(1 + εn)−). Le T1={T1n;n∈
N}, le T2={d(bn,λ
−); n∈N}, le T3={d(cn,λ
−); n∈N}, le T4=
{d(dn,λ
−); n∈N}and le T=T1∪T
2∪T
3∪T
4. We define Ua, as
ollows:
Ua, =d(a, −)
n∈N
(T1n∪d(bn,λ
−)∪d(cn,λ
−)∪d(dn,λ
−)).
By cons uc ion we ha e a∈Ua, and diam(Ua, )≤ . Le us suppose
ha diam(Ua, )< . Then o 1, 2∈] max(λ, diam(Ua, )), [ and o
u∈Ksuch ha 1<|u−a|<
2<|a−b|, he disk d(u, |a−u|−)
is a union o holes o T. By Lemma 17, he e exis s T∈T such ha
d(u, |a−u|−)=T. Mo e p ecisely, as λ<|u−a|, we ha e T∈T
1. Thus,
he annulus Γ(a, 1,
2) admi s a pa i ion Pby elemen s o T1. Le
∈] 1,
2[. By [12, P oposi ion 2.5] he e exis s an idempo en inc easing
T-sequence S, o diame e whose elemen s lie in T1. By hypo hesis we
ha e lim
n→∞ εn= 0, so his sequence is simila o a sequence SDo holes
o D. The D-beach o SDcon ains bbecause |a−b|> . Hence, since
Dis co ec ly pie ced, so is SD. Consequen ly, as Sis a T-sequence,
using Co olla y 11, we see ha SDis a T-sequence, which is absu d
because Ddoesn’ admi T-sequences wi h a no emp y D-beach. Thus
diam(Ua, )= .
We will check ha a hole o Ua, is an elemen o T. Indeed, le Tbe
a hole o Ua, . I is clea ly seen ha Tis a union o holes o T. Then
by Lemma 17, he e exis s T∈T such ha T=T. We deduce ha
he diame e s o holes o Ua, a e supe io o λand consequen ly, Ua, is
well pie ced. Besides, Ua, is closed by cons uc ion.
Nex , gi en a mono onous dis ances ( esp. weigh ed) sequence So
holes o Ua, , we may deno e by S1( esp. S2, esp. S3, esp. S4) he
subsequence which consis s o he holes o Slying in T1( esp. T2, esp. T3,
esp. T4).
We will show ha Ua, is co ec ly pie ced. Then, since Ua, is well
pie ced, we only ha e o p o e ha e e y mono onous dis ances se-
quence So holes o Ua, , o diame e ρ, ei he has an emp y Ua, -beach
o has a supe io limi -pie cing s ic ly in e io o ρ. Wi hou loss o
gene ali y we may suppose ha Sis o cen e 0. I S1is infini e, hen i
is simila o a sequence SDo holes o D.AsUa, ⊂Dand as Dis an
analoid, we see ha i Shas a no emp y Ua, -beach, hen SDand S1
a e co ec ly pie ced.
158 K. Boussa
Defini ion. A se D⊂Kis said o be ci cled i a leas one o he
ollowing s a emen s is sa isfied:
i) Dadmi s a ci cled hole.
ii) Dis pe iphe ally ci cled.
O he wise i is said unci cled.
P oposi ion 27. Le Dbe an analoid. The ollowing s a emen s a e
equi alen :
a) A(D)=H(D).
b) Dis D-admissible.
P oo : Clea ly b) implies a).
I Dis no D-admissible hen, since i is an analoid, ei he Dis no
closed o is no bounded o is no well pie ced o is ci cled.
Fi s suppose ha Dis no closed and, wi hou loss o gene ali y,
ha 0 ∈D D( esp. we suppose ha Dis unbounded). Le (an)n∈N
be a sequence in K∗such ha lim
n→∞
n
!|an|= 0. Then, such a sequence
sa isfies lim
n→∞
|an|
sn= 0 ( esp. lim
n→∞ |an|sn= 0) o all s>0. Hence we see
ha he se ies =
∞
n=0
an
xn( esp. =
∞
n=0
anxn) belongs o A(K {0})
( esp. A(K)). In pa icula he es ic ion /D o o Dbelongs o
A(D), bu i is well known ha /D is no in H(D).
I Dis no well pie ced, by [6, Theo em 19.7] H(D) is no s able by
de i a ion. Hence, acco ding o P oposi ion 22, H(D) is s ic ly included
in A(D).
Finally, suppose ha Dadmi s a ci cled hole ha we suppose (wi hou
loss o gene ali y) equal o d(0, −) ( esp. we suppose ha Dis pe iph-
e ally ci cled o diame e S). Then le (an)n∈Nbe a sequence in K∗
sa is ying
(1) (|an|
n)n∈N( esp. (|an|Sn)n∈N) is unbounded.
(2) lim
n→∞
n
!|an|= ( esp. lim
n→∞
n
!|an|=1
S).
S ic ly analy ic unc ions on p-adic analy ic open se s 159
Then such a sequence sa isfies lim
n→∞
|an|
sn=0∀s> ( esp. lim
n→∞|an|sn=0
∀s<S). Hence he se ie =
∞
n=0
an
xn( esp. =
∞
n=0
anxn) belongs o
A(K d(0, )) ( esp. A(d(0,S−))) and consequen ly i s es ic ion o D
belongs o A(D). Bu acco ding o (1), we see ha /D is no in H(D)
and his ends he p oo o P oposi ion 27.
6. Analy ic con inua ion.
In Theo em 28 we will show ha s ic ly analy ic unc ions on an
analoid o Ksa is y he p ope y o analy ic con inua ion.
Theo em 28. Le Dbe an analoid in K, ∈A(D),a∈Dand
>0.I (x)=0 o all x∈d(a, )∩D hen (x)=0 o all x∈D.
P oo : Indeed, le b∈D. We will show ha (b) = 0. By Theo em 19,
he e exis s a D-admissible Ua,b which con ains aand b. I is ob ious
ha (x) = 0 o all x∈d(a, )∩Ua,b. So, since /Ua,b ∈H(Ua,b) and
since Ua,b is analy ic, we see ha (x) = 0 o all x∈Ua,b. In pa icula ,
we ha e (b)=0.
7. The diffe en ial equa ion y= y in algeb as A(D).
Le Dbe an analoid and le ∈A(D). We deno e by E( ) he
diffe en ial equa ion y= y wi h y∈A(D) and by S( ) he K- ec o
space o solu ions h∈A(D).
Theo em 29. S( )has dimension 0o 1.
P oo : Assume ha S( ) has a no iden ically ze o solu ion g. Le
a∈Dsuch ha g(a)= 0. Le hbe ano he no iden ically ze o solu ion.
Le b∈D, hen by Theo em 19, he e exis s a D-admissible se Ua,b
con aining aand b. Fo e e y l∈A(D), le la,b be he es ic ion o l o
Ua,b. Then ga,b,ha,b a e solu ions o he equa ion y= a,byin H(Ua,b).
Bu since g(a)= 0, by [6, Theo em 55.4], we ha e h(b)=h(a)
g(a)g(b).
The e o e, i we pu λ=h(a)
g(a), we see ha h(x)=λg(x) o all x∈D.
Example. Le (an)n∈Nand (αn)n∈Nbe wo sequences in Ksa is ying
|an|<|an+1|,|αn|<|αn+1|∀n∈N, lim
n→∞|an|= 1, lim
n→∞|αn|= 2,
∞
n=0
−log |an|<+∞and
∞
n=0
(log 2 −log |αn|)<+∞. Le (ρn)n∈Nbe a
160 K. Boussa
sequence in ]0,1[ such ha lim
n→∞ρn= 0 and choose (λn)n∈Nand (µn)n∈N
wo sequences in Ksuch ha 0 <lim
n→∞
|λn|
ρn
, lim
n→∞ µn= 0 and |λn|<ρ
n
∀n∈N.
Fo e e y n∈N, we pu bn=an+λnand βn=αn+µn.
Le D=d(0,3) "
n∈N
d(an,ρ
−
n)) {αn;n∈N}{βn;n∈N}#. Ac-
co ding o [6, Lemma 4 and P oposi ion 36.5], we check ha Ddoesn’
admi T-pola sequences and he e o e is analoid.
Le =
∞
n=0
λn
(x−an)(x−bn)+
∞
n=0
µn
(x−αn)(x−βn). Since any D-
admissible se Uis well pie ced, and since lim
n→∞ λn= lim
n→∞ µn= 0, i is
easy o show ha he es ic ion o o Uis an elemen o H(U) and
consequen ly ∈A(D).
Le E( ) be he diffe en ial equa ion
y= y in A(D).
The unc ion g=
∞
n=0"x−bn
x−an#∞
n=0"x−αn
x−βn#lies in A(D) and is a solu ion
o E( ). Indeed, fi s we show ha gi en a D-admissible se U, he
p oduc hm=
m
n=0"x−bn
x−an#con e ges uni o mly on U.AsUis well
pie ced we may choose σ>0 such ha diame e o holes o Ua e lowe
bounded by σ. Since lim
n→∞ |an−bn|= 0, he e exis s N∈Nsuch ha
|an−bn|<σ∀n≥N. Then, we see ha we ha e |x−bn
x−an
|=1∀x∈U,
∀n≥N. The e o e, o m≥Nwe ha e
hm+1 −hmU=hNU$$$$
x−bm+1
x−am+1
−1$$$$U
.
So, as hNis i ially bounded on Uand as
$$$$
x−bm+1
x−am+1
−1$$$$U
=$$$$
am+1 −bm+1
x−am+1 $$$$U
≤|am+1 −bm+1|
σ
we see ha lim
m→∞ hm+1 −hmU= 0. The e o e h=
∞
n=0"x−bn
x−an#is an
analy ic elemen on any D-admissible se Uand consequen ly h∈A(D).
S ic ly analy ic unc ions on p-adic analy ic open se s 161
Simila ly we show ha l=
∞
n=0"x−αn
x−βn#is an elemen o A(D). As
g=hl,gis an elemen o he K-algeb a A(D). Bu clea ly, g/∈H(D).
Finally we check ha g
g= . By Theo em 29, he space o solu ions
S( ) has dimension 1 and hen i is he subspace gene a ed by gin A(D).
Re e ences
1. Y. Amice, Les nomb es p-adiques, P.U.F., Pa is (1975).
2. V. G. Be ko ich,“Spec al heo y and analy ic geome y o e
non a chimedean fields,” Ma hema ical Su eys and Monog aphs
33, Ame . Ma h. Soc., P o idence, RI, 1990.
3. S. Bosch, U. G ¨
un ze and R. Remme ,“Non-A chimedean
Analysis,” Sp inge -Ve lag, Be lin, Heidelbe g, New Yo k, 1984.
4. A. Escassu ,´
El´emen s analy iques e fil es pe c´es su un ensemble
in aconnexe, Ann. Ma . Pu a Appl. (4) 110 (1976), 335–352.
5. A. Escassu ,T-fil es, ensembles analy iques e ans o ma ion
de Fou ie p-adique, Ann. Ins . Fou ie (G enoble) 25(2) (1975),
45–80.
6. A. Escassu ,“Analy ic elemen s in p-adic analysis,” Wo ld Scien-
ific Publishing, Singapo e, 1995.
7. J. F esnel and M. Van-De Pu ,“G´eom´e ie analy ique igide
e applica ions,” Bi kh¨ause , Bos on-Basel-S u ga , 1981.
8. G. Ga andel, Les semi-no mes mul iplica i es su les alg`eb es
d’´el´emen s analy iques au sens de K asne , Indag. Ma h. 37(4)
(1975), 327–341.
9. B. Guennebaud, Su une no ion de spec e pou les alg`eb es no -
m´ees ul am´e iques, Th`ese, Uni e si ´e de Poi ie s (1973).
10. R. Ka lowski and P. Ull ich, A Ta e- heo e ic iew o K as-
ne ’s non-a chimedean unc ion heo y, Ma h. Ann. 289 (1991),
403–419.
11. M. K asne , P olongemen analy ique uni o me e mul i o me
dans les co ps alu´es comple s. Les endances g´eom´e iques en
alg`eb e e h´eo ie des nomb es, Cle mon -Fe and (1964), 94–141.
Cen e Na ional de la Reche che Scien ifique, 1966, (Colloques in-
e na ionaux de C.N.R.S. Pa is, 143).
12. N. Maine i, Algeb as o abs ac analy ic elemen s, in “p-adic
unc ional analysis,” Lec u e No es in Pu e and Applied Ma he-
ma ics 192, Dekke , New Yo k, 1997, pp. 281–295.
162 K. Boussa
13. Y. Mo i a, Analy ic unc ions on an open subse o P1(k), J. Reine
Angew. Ma h. 311/312 (1979), 361–383.
14. Ph. Robba, Fonc ions analy iques su les co ps alu´es ul am´e-
iques comple s. P olongemen analy ique e alg`eb es de Banach
ul am´e iques, As ´e isque 10 (1973), 109–220.
15. M. C. Sa man and A. Escassu ,T-sui es idempo en es, Bull.
Sci. Ma h. (2) 106(3) (1982), 289–303.
16. M. C. Sa man and A. Escassu , The equa ion y=ωy and
me omo phic p oduc s, in “p-adic unc ional analysis,” Lec u e
No es in Pu e and Applied Ma h. 137, Dekke , New Yo k, 1992,
pp. 157–175.
17. J. Ta e, Rigid analy ic spaces, In en . Ma h. 12 (1971), 257–289.
Labo a oi e de Ma h´ema iques Pu es
Uni e si ´e Blaise Pascal (Cle mon -Fe and)
Complexe Scien ifique des C´ezeaux
63177 Aubie e Cedex
FRANCE
e-mail: b[email p o ec ed]cle mon .
P ime a e si´o ebuda el 29 de gene de 1998,
da e a e si´o ebuda el 19 de juny de 1998
