Gevrey Expansions of Hypergeometric Integrals II
Abstract
We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.
Full text
Cas o-Jiménez e al. (2021) “Ge ey Expansions o Hype geome ic In eg als II,”
In e na ional Ma hema ics Resea ch No ices, Vol. 2021, No. 23, pp. 17823–17861
Ad ance Access Publica ion Janua y 7, 2020
h ps://doi.o g/10.1093/im n/ nz303
Ge ey Expansions o Hype geome ic In eg als II
F ancisco-Jesús Cas o-Jiménez1, Ma ía-C uz
Fe nández-Fe nández1,∗and Michel G ange 2
1Depa amen o de Álgeb a e Ins i u o de Ma emá icas-IMUS,
Uni e sidad de Se illa, A . Reina Me cedes s/n 41012 Se illa, Spain and
2Uni e si é d’Ange s, Dépa emen de Ma héma iques, LAREMA, CNRS
UMR n. 6093, 2 Bd. La oisie , 49045 Ange s, F ance
∗Co espondence o be sen o: e-mail: [email p o ec ed]
We s udy in eg al ep esen a ions o he Ge ey se ies solu ions o i egula hype geo-
me ic sys ems unde ce ain assump ions. We p o e ha , o such sys ems, any Ge ey
se ies solu ion, along a coo dina e hype plane o i s singula suppo , is he asymp o ic
expansion o a holomo phic solu ion gi en by a ca e ully chosen in eg al ep esen a ion.
1 In oduc ion
In [10] (see also [11,12]) he au ho s in oduce and s udy A-hype geome ic sys ems
and hei solu ions, gene alizing many classical hype geome ic di e en ial equa ions.
Gene al A-hype geome ic sys ems, also known as GKZ sys ems, a e ini ely gene a ed
D-modules, whe e D:=C[x]∂=C[x1,...,xn]∂1,...,∂ns ands o he complex n- h
Weyl algeb a.
Le us i s ecall some p elimina y no ions and esul s in D–module heo y.
Gi en a le D–ideal J⊆D, we conside he cyclic D–module M:=D/J. A solu ion o
Mis an elemen o a le D–module Fsuch ha P· =0, ∀P∈J. In his pape we only
conside he cases when Fis ei he he space o holomo phic unc ions o he space
Communica ed by P o . Masaki Kashiwa a
Recei ed Ap il 16, 2019; Re ised Oc obe 04, 2019; Accep ed Oc obe 10, 2019
© The Au ho (s) 2020. Published by Ox o d Uni e si y P ess. All igh s ese ed. Fo pe missions,
please e-mail: jou nals.pe [email protected].
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17824 F.-J. Cas o-Jiménez e al.
o Ge ey se ies (o o de s∈R)alongY={xn=0}a p∈Y. We ecall ha such a
Ge ey se ies is an exp ession o he o m =∞
m=0 mxm
nwhe e m= m(x1,...,xn−1)
is holomo phic a pand ∞
m=0 mxm
n/(m!)s−1is con e gen a p. The smalles possible s
(i any) so ha his la e condi ion holds is called he Ge ey index o .
On he o he hand, i u, ∈Rnsa is y u+ ∈Rn
>0one can conside he g aded
ideal (o ini ial ideal)o Jwi h espec o L=(u, ), deno ed by inL(J), which is an ideal
in he polynomial ing C[x,ξ]=C[x1,...,xn,ξ1,...,ξn], see, o example, [3, page 28]. I s
ze o se V(inL(J)) ⊆C2nis he L–cha ac e is ic a ie y o he cyclic D–module M=D/J,
see, o example, [28, De ini ion 3.1]. I F=(u, )wi h u=(0, ...,0), =(1, ...,1) hen
Ch(M):=V(inF(J)) is simply called he cha ac e is ic a ie y o M.TheD–module M
is said o be holonomic i he dimension o Ch(M)is n.Thesingula locus o Mis he
Za iski closu e o he image o Ch(M) {ξ1= ··· = ξn=0}⊆C2nby he p ojec ion
C2n−→ Cn,(x,ξ) → x. On he o he hand, se V:=(−en,en), whe e en=(0, ...,0,1),
and deno e Ls:=F+(s−1)V o s>1. The Ls-cha ac e is ic a ie y is known o be
locally cons an wi h espec o s>1 excep a a ini e se o alues called he slopes o
Malong Y, see [17]. I Mis holonomic and i has a Ge ey solu ion wi h Ge ey index
s>1alongY hen sisaslopeo Malong Y, see [18, Théo ème 2.4.2] and [23] o a mo e
gene al and s onge s a emen .
The inpu da a o a GKZ sys em is a pai (A,β) whe e βis a ec o in Cdand
A=(ak)=(a(1),...,a(n)) ∈(Zd)nisad×nma ix whose - h column is a() and
ZA:=d
k=1Za(k)=Zd.The o ic ideal IA⊂C[∂]:=C[∂1,...,∂n] is he ideal gene a ed
by he amily o binomials ∂u−∂ , whe e u, ∈Nnand Au =A (we assume 0 ∈N).
Following [10,11], he hype geome ic ideal associa ed wi h he pai (A,β) is
HA(β) :=DIA+D(E1−β1,...,Ed−βd),
whe e Ek=n
=1akx∂is he k− h Eule ope a o associa ed wi h he k- h ow o A.
The co esponding hype geome ic D-module (o A-hype geome ic sys em) is MA(β) :=
D
HA(β) .
In [11]and[1, Thm. 3.9] he au ho s p o e ha any hype geome ic sys em MA(β)
is holonomic. Mo eo e , a cha ac e iza ion o he egula i y o MA(β), in he sense o
D–module heo y [18,23], is p o ided in he se ies o pape s [16,27,28]. The holonomic
D-module MA(β) is egula i and only i he o ic ideal IAis homogeneous o he
s anda d g ading in he polynomial ing C[∂]. In pa icula he condi ion o be egula
o MA(β) is independen o he pa ame e ec o β.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17825
The dimension o he space o ge ms o holomo phic solu ions o MA(β) a ound
a gene ic poin in Cnequals d!Vol(A)i βis gene ic (see [11], [1, Co . 5.20], and [22]).
He e Ais he con ex hull in Rdo he poin s 0,a(1),...,a(n), whe e 0∈Rdis he o igin,
and Vol(A)is i s Euclidean olume. These holomo phic solu ions a e ep esen ed as
–se ies in [11] (see also [24]and[7]) when βis gene ic enough.
A. Adolphson conside s in [1, Sec. 2] in eg al ep esen a ions o solu ions o
MA(β) ha in ol e exponen ials o polynomial unc ions and app op ia e in eg a ion
cycles. In [6], A. Es e o and K. Takeuchi p o e ha he gene ic holomo phic solu-
ion spaces a e in ac comple ely desc ibed by Adolphson’s in eg al ep esen a ions
along apid decay cycles as in oduced by M. Hien in [14]and[15]. Such ype o
in eg als a e also used in [20] and gene alized in [21], whe e hey a e called Laplace
in eg als.
The slopes, see [18], o MA(β) along coo dina e subspaces a e desc ibed in
[28]. Thei co esponding i egula i y shea es and Ge ey se ies solu ions, de ined
in [23], a e s udied and desc ibed o gene ic pa ame e s βin [7] (see also [8,9]).
Mo eo e , in [4, P oposi ion 5.3 and Rema k 5.4] hese Ge ey se ies solu ions o
MA(β) a e in e p e ed as asymp o ic expansions o ce ain o i s holomo phic solu ions
unde some assump ion on he Ge ey index o he se ies, ia he so-called modi ied
A-hype geome ic sys ems in oduced in [29].
In [5], and when Ais a ow ma ix wi h posi i e in ege en ies, he au ho s
de elop a link be ween Ge ey se ies solu ions o MA(β) and holomo phic solu ions in
sec o s ollowing Adolphson’s app oach. They p o e ha any Ge ey se ies solu ion,
along he singula suppo o he sys em MA(β), is he asymp o ic expansion o a
holomo phic solu ion gi en by a ca e ully chosen in eg al ep esen a ion.
In his pape we u he de elop his link when he ma ix A=(a(1),...,a(n)) ∈
(Zd)nsa is ies wo condi ions. Since he ank o Ais assumed o be d, we may also
assume, a e a possible eo de ing o he columns, ha he 1s dcolumns o A
de e mine a (d−1)-simplex σ. We u he assume ha Asa is ies he ollowing wo
condi ions (see Assump ion 4.1): (1) he poin s a(d+1),...,a(n−1)belong o he in e io
o he con ex hull σo σand he o igin; and (2) he poin a(n)is no in σand belongs
o he open posi i e cone o σ.Figu e1shows an example o an allowed column se
con igu a ion o a 2 ×5ma ixA, whe e σis he iangle.
Unde hese wo condi ions we ha e ha Y={xn=0}is an i educible
componen o he singula locus o MA(β) [1, Sec. 3], he e is only one slope o MA(β)
along Y[28] and, i βis gene ic enough, he dimension o he space o Ge ey se ies
solu ions o MA(β) along Yis d!Vol(σ)[7].
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17826 F.-J. Cas o-Jiménez e al.
Fig. 1.
We p o e in Theo em 4.3 ha o gene ic β∈Cd, he space o Ge ey se ies
solu ions o MA(β), along he hype plane Y, has a basis gi en by asymp o ic expansions
o holomo phic solu ions o MA(β) desc ibed by Adolphon’s in eg al ep esen a ions.
These in eg als a e solu ions o ype
IC(β;x)=IC(β;x1,...,xn):=C
−β−1exp n
=1
x a()d ,
whe e =( 1,..., d),d =d 1···d dand C uns o e a ini e se o cycles on he
uni e sal co e ing o (C∗)d. These a e Bo el–Moo e cycles o cycles wi h closed suppo
on he uni e sal co e ing o (C∗)d, a no ion o which we e e o [25, II,5.3]). Mo eo e ,
we p o e in Theo em 5.8 ha hese cycles can be eplaced by a se o apid decay
homology cycles in he sense o [15].
He e is a summa y o he con en o his pape . In Sec ion 2we conside a gene al
ma ix Aas be o e bu no necessa ily sa is ying p e ious condi ions (1) and (2) (see
Assump ion 4.1). Following a cons uc ion in [13, Sec. 4.4], we desc ibe cycles Cp,δin
he uni e sal co e ing o (C∗)d, depending on a gi en poin x∈Cn.We ixamaximal
simplex σ⊂{1, ...,n}, ha is, he se {a(k)|k∈σ}isabasiso Rd. Then his cycle
depends only on xσ:=(xk)k∈σ, and on ec o s p∈Zσand δ∈Rσwi h componen s δk
sa is ying |δk|<1/2. In Sec ion 2.3 we gi e a su icien condi ion o he in eg and
o Ip,δ(β;x):=ICp,δ(β;x) o ha e mode a e g ow h along Cp,δ. This is a s ep owa ds
su icien condi ions o con e gence o Ip,δ(β;x) ha a e de eloped in Sec ion 3.
In Sec ion 3, we pe o m he app op ia e o ic change o a iables in he
uni e sal co e ing o (C∗)d, like in [13], which educes he desc ip ion o asymp o ic
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17827
expansions o he in eg als Ip,δ(β;x) o he s udy o in eg als o ype
Fp,δ(β;y):=Dp,δ
−β−1exp ⎛
⎝ 1+···+ d+
n
j=d+1
yj a(j)⎞
⎠d ,
whe e he cycle Dp,δis he image o Cp,δunde he change o a iables. The new
in eg al Fp,δ(β;y)looks like a pa icula case o Ip,δ(β;x), wi h he 1s d×dsubma ix
(a(1),...,a(d)) equal o he iden i y ma ix. Howe e , he ma ix A=(a(1),...,a(n)) is
now allowed o ha e a ional non in ege coe icien s. The c ucial poin o con e gence
s a emen s is a condi ion o apid decay a in ini y, see inequali y (3.10). We p o e ha ,
unde some condi ions, he in eg al Fp,δ(β;y)is absolu ely con e gen when βk<0 o
k∈σand y∈(C∗)n−d; see Lemma a 3.1 and 3.2.
Sec ion 4con ains some o he main esul s o his pape . We assume ha he
ma ix Ade ined in Sec ion 3sa is ies mo e condi ions in Assump ion 4.4, deduced om
condi ions (1) and (2) in Assump ion 4.1 al eady conside ed o he o iginal ma ix. Fi s
we p o e ha he condi ions o con e gence in Lemma 4.5, can be ob ained in p ac ice
o e e y y∈Cn−dwi h yn= 0.
We ix p∈Zdand δ∈Rdonce o all and we omi hese subindexes in ou
o mulas. As a s ep owa ds p e iously men ioned Theo em 4.3, we p o e in Theo em 4.7
ha i β<0, he e is an asymp o ic expansion wi h espec o he a iable ynin some
sec o in C∗:
F(β;y)∼
yn→0
m∈N
A(β;m,y)ym
n
m!, (1.1)
whe e y=(yd+1,...,yn−1)and
A(β;m,y):=Dp,δ
−β−1+ma(n)exp ⎛
⎝ 1+···+ d+
n−1
j=d+1
yj a(j)⎞
⎠d .
Assump ion 4.4 plays an essen ial ole in he p oo o his esul . Wi hou
assump ion (1), we migh need o impose u he condi ions on he a gumen s o y,
e.g. condi ions (3.5)j o all j, in o de o gua an ee he con e gence o F(β;y). Wi hou
condi ion (2), he e ex a(n)could ha e nega i e componen s and he in eg als de ining
he coe icien s A(β;m,y)would ail o be con e gen o mla ge enough.
Then we p o e in Lemma 4.9 ha F(β;y)admi s a me omo phic con inua ion
F(β;y), wi h espec o he a iable β, wi h poles a mos in a coun able locally
ini e union o hype planes Pin Cd. The p oo o his lemma uses ha he poin s
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17828 F.-J. Cas o-Jiménez e al.
a(d+1),...,a(n)belong o d
k=1R>0a(k)=Rd
>0, which ollows om condi ions (1) and
(2). The se Pis con ained in he se o so-called esonan pa ame e s o A[12,2.9]and
i is explici ly desc ibed in e ms o he columns o A.Wealsop o einLemma4.10
ha , o any ixed pa ame e β∈ P, he me omo phic con inua ion
F(β;y)admi s an
asymp o ic expansion along yn=0 and ha he coe icien s
A(β;m,y)o his expansion
a e he analy ic con inua ion o he p e iously in oduced A(β;m,y).
In Sec ion 5we p o e ha when β<0andβis su icien ly gene al, he
in eg als F(β;y)a e in ac equal o in eg als o e apid decay cycles in he sense o
[15] (see Theo em 5.3). The s a emen s in ol ing Bo el–Moo e cycles a e weake because
he analy ic con inua ions a e no exp essed by in eg al along cycles when βk>0 o
some k. Ano he eason is ha hey a e no cycles in he sui able homology adap ed o
he p oblem, like o Hien’s apid decay homology. The no ion o apid decay cycles is
explained in Sec ion 5.1. Sec ion 5.2 is de o ed o he cons uc ion o apid decay cycles.
We s a om a p oduc o Hankel con ou s, along which he hype geome ic in eg als
a e g ossly di e gen , bu hen we build a e ined owa ds in ini y e sion o his
p oduc along which con e gen in eg als a e ob ained. These in eg als in Sec ion 5a e
also de ined when βk≥0 o some kand hey a e s ill solu ions o MA(β). In Sec ion 5.2
we p o e, by using Sec ion 4, ha hese in eg als admi asymp o ic expansions as
Ge ey se ies solu ions o MA(β) o non esonan βin Cd.
2 P oduc s o Lines o Rapid Decay
2.1 No a ions
Le us sligh ly change ou no a ion used in he in oduc ion and le us s a wi h a pai
(B,γ), whe e B:=(b(1),...,b(n))∈(Zd)nisad×nma ix, desc ibed as a lis o columns
such ha ZB:=Zb(1)+···+Zb(n)=Zdand whe e γis a pa ame e ec o in Cd.We
a e conce ned wi h in eg als:
IC(γ ;x)=IC(γ ;x1,...,xn):=C
−γ−1exp n
=1
x b()d ,
whe e 1=(1, ...,1)∈Ndand Cis a sui able cycle.
To make p ecise his de ini ion le us speci y some con en ions and no a ions.
As al eady men ioned, Cis a cycle on he uni e sal co e ing (
C∗)do (C∗)d. We iden i y
(
C∗)dwi h Cdo wi h Rd
>0×Rdand w i e z=(log +√−1θ) o ( ,θ), espec i ely, o
he coo dina es on (
C∗)dwi h θka b anch o a g k k=exp(zk),and k=| k|. We se , o
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17829
any ec o ∈Cd, =d
k=1 k
k. This is a mul i alued monomial, namely he unc ion
on he uni e sal co e ing:
exp z, =exp d
k=1
k(log k+√−1θk),
whe e we se , gi en wo ec o s u, ∈Cd,u, =d
k=1uk k.
2.2 Desc ip ion o cycles o apid decay a in ini y
I τ⊂{1, ...,n}, we deno e by Bτ he ma ix whose columns a e b(j)wi h j∈τand by τ
he complemen o τin {1, ...,n}.
Recall ha a subse σ⊂{1, ...,n}is called a maximal simplex o Bi he
columns {b(k),k∈σ} o m a basis o Rd. Such a maximal simplex σis also called a
base in [11, Sec. 1.1]. We o en iden i y he se σwi h he se o columns {b(k),k∈σ}.
We ix a maximal simplex σ o Band ake x∈Cnsuch ha xk= 0 o all k∈σ.
We also ix p=(pk)k∈σ∈Zσ≃Zd,δ=(δk)k∈σ∈Rσ≃Rdsuch ha |δk|<1
2 o all k∈σ.
We deno e by Cp,δ he cycle in he space (
C∗)ddesc ibed by he ollowing condi ion on
he a gumen θ:=a g o ∈(C∗)d(i.e. θ:=(a g 1,..., a g d)):
a g(xk b(k))=a g xk+b(k),θ=(1+δk+2pk)π o allk∈σ. (2.1)
Rema k 2.1. The cycle Cp,δdepends on xσ:=(xk)k∈σ∈(C∗)σ≃(C∗)dandalsoona
choice o i s a gumen . Howe e , a change in his choice yields only a eindexa ion by p
o he unchanged se o hese cycles. Fo ha eason in all ou s a emen s we s ick on
xσ∈(C∗)σwi hou passing o he uni e sal co e ing o (C∗)σ.
F om now on we will deno e Ip,δ(γ ;x)=ICp,δ(γ ;x). The cycles Cp,δa easligh ly
modi ied e sion o cycles conside ed in [13, Sec. 4.4].
Le us se :=π
2,3π
2+2πZ. The equali y (2.1) can be globally ew i en using
ma ix no a ion:
a g xσ+ Bσθ=(1+δ+2p)π ∈σ. (2.2)
The e is a unique solu ion θo he p e ious equa ion
θ=( Bσ)−1−a g xσ+(1+δ+2p)π(2.3)
so ha Cp,δis he ca esian p oduc o dopen hal –lines.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17830 F.-J. Cas o-Jiménez e al.
Gi en p,p∈Zd,le θ=a g ,θ=a g be he co esponding unique solu ions
o equa ion (2.2).
I ( Bσ)−1(p−p)∈Zd hen θ−θ∈2πZdand he p ojec ions o he wo cycles Cp,δ
and Cp,δon (C∗)da e he same. We check ha he con e gence o he wo in eg als along
he cycles Cp,δand Cp,δa e hen equi alen o each o he and, mo eo e , he in eg al
solu ions di e only by a cons an ac o :
Ip,δ(γ ;x)=Cp,δ
−γ−1exp n
=1
x b()d =e−2π√−1(d
k=1mkγk)Ip,δ(γ ;x)
o some mk∈Z,k=1, ...,d.
When p a ies in a se o ep esen a i es o Zd
Z Bσ, we will see ha he
con e gence o he in eg al Ip,δ(γ ;x)depends on δ(see Rema k 2.2 and Lemma 4.5).
Howe e , choosing in each such class an app op ia e δ, we can ind, as a consequence o
ou main esul and unde some condi ions (see Assump ion 4.1), [Zd:Z Bσ]=|de Bσ|
many in eg al solu ions Ip,δ(γ ;x)which a e linea ly independen (see Theo em 4.3).
We will see in he p oo o Lemma 3.2, a e he change o a iables de ined in
Sec ion 3, ha he cycles Cp,δa e o apid decay a in ini y.
2.3 Su icien condi ions o mode a e g ow h
Su icien condi ions o he con e gence o he in eg al Ip,δ(γ ;x)a ede ailedin he
nex sec ion (see Lemma 3.2 and Rema k 3.6). As a p elimina y s ep le us look he e a a
condi ion o bounding he exponen ial e m in ha in eg al; le us no ice ha condi ion
(2.1) implies ha (xk b(k))<0alongCp,δ o any xk∈C∗,k∈σ. I we addi ionally could
ensu e ha
a g(xj b(j))∈ o all j∈σsuch ha xj= 0 (2.4)
(see Rema k 2.2 below) hen he a gumen o he exponen ial has nega i e eal pa along
Cp,δ; hence, he absolu e alue o he exponen ial e m in he in eg al Ip,δ(γ ;x)is bounded
by 1. Then i we ake in o accoun he e m −γ−1, he in eg and o Ip,δ(γ ;x)has mode a e
g ow h along Cp,δ.
Rema k 2.2. Le us no ice ha condi ion (2.3) de e mines a unique cycle Cp,δ o a
gi en pand δ. I is no clea ha o gi en x∈(C∗)σ×Cσand p∈Zσone can always
choose δ∈Rσ o his cycle o sa is y condi ions (2.2)and(2.4). I is he e o e in e es ing
o weaken hese condi ions by keeping only he signi ican ones. In Lemma 3.2 and
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17831
Rema k 3.6, comple ed by Rema k 3.3, we do his wi h a educed e sion o he a iables
x enamed y.
We will see in he nex sec ion ha condi ions (2.2)and(2.4) a e su icien
con e gence condi ions o he in eg als Ip,δ(γ ;x)when combined wi h a condi ion on
he pa ame e γ. A e an app op ia e change o a iables we can in e p e hem as a
condi ion o apid decay a in ini y, see he p oo o Lemma 3.2.
We no ice ha Cp,δis a Bo el–Moo e cycle in (
C∗)dbu no in gene al a apid
decay cycle in he sense o [14], see Rema k 3.5.
Howe e , we shall p o e in Sec ion 5 ha he in eg al along Cp,δis equal o
an in eg al along a apid decay cycle (see Theo em 5.3) unde Assump ion 4.4,and o
alues o γ ha gua an ee con e gence. Ou esul can be hen in e p e ed in he ame
o [6, Th. 4.5].
3 A Change o Va iables and Explici Calcula ions
We will assume o simplici y, a e a possible eo de ing o he a iables, ha he
maximal simplex σis {1, ...,d}.Le us ixx∈(C∗)d×Cn−d, and an a gumen o all xk
wi h k∈σ. We conside he ini e o one co e ing (C∗)d→(C∗)do deg ee de Bσ,gi en
by he o mula:
sk=xk b(k) o k∈σ.
We hink o i as a ( ami ied) o ic change o a iables. We ix a b anch o log xσand we
conside he bijec i e change o a iables on he uni e sal co e ing (
C∗)d≃Cd,gi enby
log sk−log xk=log ·b(k).
F ac ional powe s like x
σwi h ∈Qdha e he na u al meaning x
σ=exp(log xσ·
), and he in e se mapping on (
C∗)dcan be ead as ollows using hese ac ional
powe s:
k=s
xσB−1
σe(k)
o k∈σ,
whe e s
xσis he ec o wi h coo dina es sk/xkand (e(k))k∈σis he s anda d basis o Zd.
The image Dp,δo he cycle Cp,δdesc ibed in Sec ion 2, is de e mined by he condi ions:
a g sk=(1+δk+2pk)π o all k∈σ. (3.1)
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17838 F.-J. Cas o-Jiménez e al.
Rema k 4.6. Le us ix p,δand a b anch αno a g yn,0 sa is ying condi ion (??)n.This
de ines he subse R⊂Ro allowed a gumen s a g yn.Le Sp,δ⊂C∗be he sec o
de ined by a g yn∈R0 he connec ed componen o αnin R. Like in Rema k 3.3,Sp,δis
independen o he choice o αn.
Theo em 4.7. I βk<0 o all k=1, ...,d, hen o any gi en yn,0 ∈C∗ he e is an
asymp o ic expansion wi h espec o he a iable ynin he open sec o Sp,δ:
Fp,δ(β;y)∼
yn→0
m∈N
Ap,δ(β;m,y)ym
n
m!,
whe e y=(yd+1,...,yn−1)and
Ap,δ(β;m,y):=Dp,δ
−β−1+ma(n)exp ⎛
⎝ 1+···+ d+
n−1
j=d+1
yj a(j)⎞
⎠d .
P oo . We ha e o p o e ha o any in ege N>0 he e exis s KN=KN(β,y)>0
such ha Fp,δ(β;y)−
N−1
m=0
Ap,δ(β;m,y)ym
n
m!≤KN|yn|N
holds o e e y yn∈Sp,δ.
Le
N(z):=ez−
N−1
m=0
zm
m!
o z∈C.Thenweha e
|N(z)|≤|z|N
N! o all zsuch ha z<0.
Recall ha by he assump ion on δwe ha e (yn a(n))<0when ∈Dp,δsince
yn∈Sp,δ.Thus,weha e
Fp,δ(β;y)−
N−1
m=0
Ap,δ(β;m,y)ym
n
m!=yN
nQp,δ(β;y,N),
whe e
Qp,δ(β;y,N)=Dp,δ
a(n)N−β−1exp ⎛
⎝ 1+···+ d+
n−1
j=d+1
yj a(j)⎞
⎠N(yn a(n))
(yn a(n))Nd .
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17839
The absolu e alue o he in eg and in Qp,δ(β;y,N)is bounded by he unc ion
1
N! a(n)N−β−1exp ⎛
⎝ 1+···+ d+
n−1
j=d+1
yj a(j)⎞
⎠,
which is independen o ynand in eg able o e Dp,δby Lemma 3.2 ( ha can be applied o
he subma ix o Ade ined by i s 1s n−1 columns because o Assump ion 4.4). No ice
ha we use he e ha (β −a(n)N)<0 o all N>0sincea(n)does no ha e nega i e
coo dina es.
Thus, he e exis s KN=KN(β,y)>0 such ha Qp,δ(β;y,N)≤KN. This inishes
he p oo .
Rema k 4.8. No ice ha Ap,δ(β;m,y)=Fp,δ(β −ma(n);y) o he subma ix o A
de ined by i s 1s n−1 columns. In pa icula i is analy ic wi h espec o (β,y)by
Rema k 3.4.
We ex end Theo em 4.7 o nonnega i e alues o βkin Sec ion 4.2.
4.2 Analy ic con inua ion wi h espec o β
In his sec ion we ocus on he analy ic dependency o F(β;y)=Fp,δ(β;y)on β.Le us
ake yn,0 ∈C∗and p∈Zd. We choose δas in Lemma 4.5 and we omi p,δin he emainde
o his subsec ion. We assume now ha ybelongs o Cn−d−1×Sp,δ, whe e he sec o Sp,δ
is de ined in Rema k 4.6.
The in eg al F(β;y)is a solu ion o he educed GG-sys em (see [13]):
βkF(β;y)=
n
=d+1
a()kyF(β −a();y)+F(β −e(k);y) o k=1, ...,d(4.1)
F(β −a();y)=∂F
∂y
(β;y) o =d+1, ...,n. (4.2)
Lemma 4.9. The unc ion F(β;y)admi s a me omo phic con inua ion wi h espec o
β, deno ed by
F(β;y), wi h poles a mos along he coun able locally ini e union o
hype planes
P:=
d
k=1{β∈Cd|βk∈πk(NA)},
whe e NA=Na(1)+···+Na(n)and πk:Qd→Qdeno es he p ojec ion o he k- h
coo dina e.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17840 F.-J. Cas o-Jiménez e al.
P oo . The ini ial domain o analy ici y o F(β;y)is de ined by βk<0 o all
k=1, ...,d. Le us ix condi ions βk<0 o k=2, ...,dand ex end he domain
o analy ici y in he coo dina e β1using equa ion (4.1)1as ollows. The unc ions
F(β −a();y) o =d+1, ...,nand F(β −e(1);y)a e analy ic o β1<
a1:=
min{a()1,1}and hence i ollows om equa ion (4.1)1 ha F(β;y)is me omo phic in
β1<
a1wi h a mos a pole in β1=0.
In he gene al induc i e s ep o he a iable β1, we assume ha F(β;y)is
me omo phic in he hal -space β1<(q−1)
a1. Then, on he domain de ined by
β1<q
a1, he igh -hand side o (4.1)1is me omo phic, wi h poles o ype β1=c+1, o
β1=c+a()1, whe e β1=c uns o e all he poles o F(β;y). We ob ain ha F(β;y)
is also me omo phic in he same domain adding hese new poles o hose al eady
ound. Thus, by induc ion, we ge ha F(β;y)is also me omo phic o β1∈Cand
βk<0 o k=2, ...,dwi h poles a mos along β1=n
=d+1ma()1+m, o all
md+1,...,mn,m∈N. By an analogous a gumen in k=2, ...,dwe ge he esul .
No ice ha he equa ions (4.2) a e hen sa is ied by
F(β;y)by analy ic con inu-
a ion on U:=Cd P.
Lemma 4.10. Fo any ixed β∈,
F(β;y)admi s an asymp o ic expansion along yn=
0inSp,δ. Fu he mo e, he coe icien s
A(β;m,y)o his expansion a e analy ic wi h
espec o β∈U. Hence, hey a e analy ic con inua ions o he coe icien s A(β;m,y)
desc ibed in Theo em 4.7.
P oo . I ollows om an induc ion s a ing om Theo em 4.7 and pa allel o he
one used in he p oo o Lemma 4.9 ha o any ixed β∈,
F(β;y)admi s asymp o ic
expansions along yn=0inSp,δ. By cons uc ion, hese analy ic con inua ions sa is y
equa ion (4.1), o any β∈U. This implies ha he coe icien s
A(β;m,y)o hese
expansions sa is y he ollowing equa ions o k=1, ...,d:
βk
A(β;m,y)=
A(β −e(k);m,y)+
n−1
=d+1
a()ky
A(β −a();m,y)
+ma(n)k
A(β −a(n);m−1, y).
(4.3)
Again by an induc ion like in Lemma 4.9,using(4.3) and Rema k 4.8,
A(β;m,y)
is analy ic wi h espec o βand y, hence as a unc ion o βi is an analy ic con inua ion
o Uo A(β;m,y).
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17841
We ha e p o ed he ollowing heo em ha implies he las sen ence in
Theo em 4.3 when we e u n o he in eg als IC(β;y).
Theo em 4.11. The e is an asymp o ic expansion along yn=0, in an app op ia e open
sec o Sp,δa ound any hal -line R>0·yn,0 ⊂C∗:
Fp,δ(β;y)∼
yn→0
m∈N
Ap,δ(β;m,y)ym
n
m!,
whe e
Ap,δ(β;m,y)is he analy ic con inua ion o Ap,δ(β;m,y) o β∈U.
4.3 Pa ame iza ions
We go on wo king wi h he educed o m o he in eg al desc ibed in (3.2–3.4), and we
s udy in eg als o he o m
Fp,δ(β;y)=Dp,δ
−β−1exp ⎛
⎝ 1+···+ d+
n
j=d+1
yj a(j)⎞
⎠d .
Lemma 4.12. I β<0, hen
A0
p,δ(β) :=Dp,δ
−β−1exp( 1+···+ d)d =e√−1π2p+1,−β(−β),
whe e (−β) :=d
k=1(−βk).
P oo . The in eg and −β−1exp( 1+···+ d)d is o apid decay a in ini y in he p oduc
o dsec o s de ined by he condi ion:
a g( k)∈[(1+min{0, δk}+2pk)π,(1+max{0, δk}+2pk)π], k∈σ.
Thus, since his p oduc o sec o s con ain Dp,0 and Dp,δ, we know by elemen a y
conside a ions in one complex a iable, ha A0
p,δ(β) does no depend on δk∈]−1
2,1
2[and
so A0
p,0(β) =A0
p,δ(β).
We pa ame ize Dp,0 by k=ρke√−1π(2pk+1)=−ρkwi h ρk∈]0, +∞),and he
esul ollows di ec ly om he exp ession ha we ob ain
A0
p,0(β) =]0,+∞)dexp(√−1π2p+1,−β)ρ−β−1exp(−ρ1−···−ρd)dρ.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17842 F.-J. Cas o-Jiménez e al.
Since A0
p,δ(β) does no depend on δ, om now on we d op δand se A0
p(β) :=
A0
p,0(β) =A0
p,δ(β).
We no ice ha Fp,δ(β;y)is locally cons an wi h espec o δby a simila
homo opy a gumen . Howe e , he dependency on δo Fp,δ(β;y)mus be kep because
he a gumen by homo opy wo ks only o small pe u ba ions o δ. This does no allow
a educ ion o δ o ze o. Le us now make he analy ic con inua ion o he coe icien s
o he asymp o ic expansion desc ibed in Theo em 4.7 mo e p ecise by de eloping hem
wi h espec o y.
Lemma 4.13. The coe icien s o he asymp o ic expansion desc ibed in Theo em 4.7
a e analy ic unc ions o he a iables ywi h he ollowing powe se ies de elopmen :
Ap,δ(β;m,y)=
m∈Nn−d−1
A0
p⎛
⎝β−ma(n)−
n−1
j=d+1
mja(j)⎞
⎠ym
m!. (4.4)
Fu he mo e, his expansion is s ill alid o he me omo phic con inua ion o
Ap,δ(β;m,y) ound in Lemma 4.10 and he me omo phic con inua ion o A0
p(β) deduced
om Lemma 4.12.
P oo . Recall ha , when β<0 he coe icien we conside has he o m
Ap,δ(β;m,y)=Dp,δ
ϕ(β;y; )d
wi h
ϕ(β;y; )= −β−1+ma(n)exp ⎛
⎝ 1+···+ d+
n−1
j=d+1
yj a(j)⎞
⎠.
We se | k|=ρk o k=1, ...,dand we pa ame ize Dp,δby ρ∈Rd
>0.We ixa
polydisc Q={y||yj|<Rj,j=d+1, ...,n−1}⊂Cn−1−d.
The unc ion ϕ(β;y; )is holomo phic wi h espec o y∈Cn−d−1. By he same
a gumen as in he p oo o Lemma 3.2 and inequali y (3.10), he in eg and ϕ(β;y; )d
is domina ed, ia he pa ame iza ion k=e(1+δk+2pk)√−1πρkand up o a cons an
ac o , by
ρ−β+ma(n)−1exp(C−c(ρ1+···+ρd))dρ
o some cons an s C,c∈R>0. These cons an s depend only on Qbu no on y∈Qby
Rema k 3.4 applied o Ap,δ(β;m,y)ins ead o Fp,δ(β;y).
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17843
Fo each j=d+1, ...,n−1, he in eg al
Dp,δ
∂ϕ(β;y; )
∂yj
d
has an exp ession simila o he one o Ap,δ(β;m,y),wi hβ eplaced by β−a(j).By he
same a gumen as o ϕ, he in eg and ∂ϕ(β;y; )
∂yjd is domina ed, up o a cons an ac o ,
by
ρ−β+ma(n)+a(j)−1exp(Cj−cj(ρ1+···+ρd))dρ
o some cons an s Cj,cj∈R>0, independen o yin he polydisk Q.
By Lebesgue’s heo em on domina ed con e gence o in eg als, hese conside -
a ions p o e ha Ap,δ(β;m,y)is holomo phic wi h espec o yand ha
∂Ap,δ(β;m,y)
∂yj=Dp,δ
∂ϕ(β;y; )
∂yj
d
o all j=d+1, ...,n−1. I we i e a e he a gumen we ob ain an exp ession o he
pa ial de i a i es o Ap,δ, up o any o de m=(md+1,...,mn−1):
∂|m|Ap,δ(β;m,y)
∂md+1yd+1···∂mn−1yn−1=Dp,δ
∂|m|ϕ(β;y; )
∂md+1yd+1···∂mn−1yn−1
d .
Se ing y=0 in his las exp ession gi es he coe icien s o he Taylo expansion o
Ap,δ(β;m,y)wi h espec o ya he o igin. This p o es he equali y (4.4)whenβ<0.
The las claim o his lemma ollows om he explici calcula ion in Lemma 4.12
om which we see ha he coe icien o ym
m!is equal o
A0
p⎛
⎝β−ma(n)−
n−1
j=d+1
mja(j)⎞
⎠=
e√−1π2p+1,−β+ma(n)+n−1
j=d+1mja(j)⎛
⎝−β+ma(n)+
n−1
j=d+1
mja(j)⎞
⎠.
By he s anda d p ope ies o he - unc ion, his coe icien admi s a me o-
mo phic con inua ion wi h espec o β, wi h poles along a subse o Pde ined in
Lemma 4.9.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17844 F.-J. Cas o-Jiménez e al.
When β∈Cd P he igh -hand side o (4.4) is s ill de ined and yields a
con e gen powe se ies de ined o all y∈Cn−d−1because o he condi ions |a(j)|<1
o j=d+1, ...,n−1, and s anda d es ima es on - unc ions (see, e.g., [7, Lemma 3.8]).
The e o e, i is an analy ic con inua ion o he powe se ies de ined o β<0.
Theequali y(4.4) ollows e e ywhe e in Cd Pwi h he me omo phic con inua ion o
Ap,δ(β;m,y)on he le -hand side, de ined in Lemma 4.10.
Rema k 4.14. No ice ha as a consequence o Lemma 4.13 he unc ion Ap,δ(β;m,y)
does no depend on δ.
4.4 Space o asymp o ic expansions and Ge ey se ies
In his sec ion we inish he p oo o Theo em 4.3. Fo any k∈Nn−d, le us se
k:={k+m=(kd+1+md+1,...,kn+mn)∈Nn−d:Aσm∈Zd}
and de ine
Sk(β;y):=
k+m∈k
e|Aσ(k+m)|π√−1−β+Aσ(k+m))yk+m
(k+m)!.
No ice ha he coe icien s o he se ies Ska e me omo phic wi h espec o
β∈Cdwi h a mos simple poles along each hype plane in P. In pa icula , i β/∈Pall
hese se ies a e well-de ined nonze o powe se ies wi h suppo equal o ksince he
Gamma unc ion does no ha e any ze o. I can be p o ed by using s anda d es ima es o
Gamma unc ions ha hese se ies a e Ge ey along yn=0 wi h Ge ey index |a(n)|>1.
Le ⊆Nn−dbe a se o ca dinali y [ZA:ZAσ] such ha
{Aσk+ZAσ:k∈}=ZA/ZAσ=ZA/Zd.
We no ice ha he exis ence o such ⊆Nn−d ollows om [7, Lemma 3.2]. I is
clea ha G={Sk(β;y):k∈}is a linea ly independen se because he se ies Sk
ha e pai wise disjoin suppo s k. Using Theo em 4.7, Lemma 4.12, and Lemma 4.13,
we ha e
Fp,δ(β;y)∼
yn→0
qn∈N
Ap,δ(β;qn,y)yqn
n
qn!
=
q∈Nn−d
A0
p(β −Aσq)yq
q!=
q∈Nn−d
e√−1π1+2p,−β+Aσq−β+Aσqyq
q!
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17845
=e√−1π1+2p,−β
k∈
k+m∈k
e√−1π1+2p,Aσ(k+m)−β+Aσ(k+m)yk+m
(k+m)!
=e√−1π1+2p,−β
k∈
e√−1π1+2p,Aσk
k+m∈k
e√−1π1,Aσm
−β+Aσ(k+m)yk+m
(k+m)!
=e√−1π1+2p,−β
k∈
e√−1π2p,AσkSk(β;y).
No ice ha p e ious powe se ies is o mal wi h espec o yn, wi h con e gen
coe icien s. Mo e p ecisely, i is a Ge ey se ies along yn=0 wi h Ge ey index
|a(n)|>1. We no ice also ha β<0 implies ha (β −Aσq)<0 o all q∈Nn−d,by
using Assump ion (4.4), which gua an ees he con e gence o all he in eg als in ol ed
in Sec ion 4.3. By he las claim in Lemma 4.13, his calcula ion is alid e e ywhe e
in he domain o analy ic con inua ion Cd P, since he a gumen applies also o he
coe icien s o he se ies Sk(β;y).
The ma ix o coe icien s o he se ies Sk(β;y)in he asymp o ic expansions o
he unc ions
e√−1π2p+1,βFp,δ(β;y)
is (e√−1π2p,Aσk)k,p, whe e k a ies in .I p a ies in an app op ia e se o [ZA:Zd]
elemen s, his ma ix is squa e and in e ible. Indeed, we ha e ZA/ZAσ=ZA/Zd≃
Zd/ZM, whe e Mis he ma ix o coo dina es o he canonical basis o Zdwi h espec
o a basis o ZA/Zd. Thus, he ma ix (e√−1π2p,Aσk)k,pis in e ible by [19, P oposi ion
6.3], i p uns in a se o ep esen a i es o he quo ien Zd/Z M. In pa icula , i β/∈P
he se o holomo phic unc ions Fp,δ(β;y), whe e p a ies in his se o ep esen a i es,
is a linea ly independen se and any Ge ey se ies along yn=0 in he space gene a ed
by he se ies {Sk(β;y):k∈}is an asymp o ic expansion o a linea combina ion o
he in eg als Fp,δ(β;y).
Now i we s a om he ma ix Bin Sec ion 2and we apply he abo e esul s
wi h he ma ix A=B−1
σB=(I,B−1
σBσ)and he pa ame e β=B−1
σγ, we ob ain a simila
s a emen o he in eg als IC(γ ;x)using (3.3)and(3.4) i we se yj=xjx−a(j)
σ o all
j=d+1, ...,n,o y=xσx−B−1
σBσ
σ. P ecisely, Mcan be chosen o be Bσ. We ge ha
xB−1
σγ
σ·Gis a linea ly independen se o Ge ey se ies solu ions o MB(γ ) along xn=0
wi h Ge ey index |a(n)|=|B−1
σb(n)|>1i β/∈P. Again his ans o ma ion in ol es a
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17846 F.-J. Cas o-Jiménez e al.
choice o a g xσ, bu by Rema k 2.1 and (3.3), a change in his choice does no modi y he
basis Gexcep o cons an ac o s.
I is enough o p o e ha he dimension o he space o Ge ey se ies solu ions
o MB(γ ) along xn=0isa mos equal o||=[Zd:ZBσ]whenβ<0. To his end,
no ice i s ha , i
=∞
m=0
m(x1,...,xn−1)xm
n
is a Ge ey se ies belonging o his space, hen he ini ial pa o wi h espec o
he weigh ec o w=(0, ...,0,1)∈Rnhas he o m inw( )= m(x1,...,xn−1)xm
n o
some m≥0 and i is hence a holomo phic unc ion. Thus, by he same a gumen as
in he p oo o [27, Th. 2.5.5], i is a (holomo phic) solu ion o in(−w,w)(HB(γ )).This
las ideal is he ini ial ideal wi h espec o wo he hype geome ic ideal associa ed
wi h (B,γ) (see [27, p. 4]). In pa icula , he dimension o he space o Ge ey solu ions
is a mos equal o he ank o in(−w,w)(HB(γ )), because one can choose a basis o
Ge ey solu ions o MB(γ ) such ha hei ini ial pa s a e also linea ly independen
(see [27, P oposi ion 2.5.7]).
On he o he hand, by using [27, Lemma 2.1.6] o (u, )=(0,1)and (u, )=
(−w,w), we ha e ha he ini ial ideal o in(−w,w)(HB(γ )) wi h espec o (0,1)is
in(0,1)(in(−w,w)(HB(γ ))) =inL(HB(γ ))
o L=(−w,w)+(0,1)wi h >0 small enough.
Thus, by [28, Th. 4.21, Rk. 4.23, and Th. 4.28] o L=(−w,w)+(0,1)and
Assump ion 4.1, we ha e ha he holonomic ank o in(−w,w)(HB(γ )) equals ||i γis
no ank–jumping o B(i.e., i ank(MB(γ )) =d!Vol(B)), a condi ion ha is weake
han β=(B−1
σγ) < 0by[1, Th. 5.15] (see also [27, Co . 4.5.3]). This inishes he p oo
o Theo em 4.3.
Rema k 4.15. No ice ha he p oo o Theo em 4.3 shows ha he cons uc ed se
o Ge ey se ies solu ions xB−1
σγ
σ·Gis s ill a basis o he space o Ge ey solu ions o
MB(γ ) along xn=0whenγis no ank-jumping and β=B−1
σγ/∈P, whe e Pis de ined
in Lemma 4.9. We do no know i unde Assump ion 4.1 he condi ion o γbeing ank-
jumping implies β∈P. Howe e , i is ue ha i γis ank-jumping hen i is semi-
esonan [1]. In pa icula , unde Assump ion 4.1,γis semi- esonan o Bi and only
i β∈P:=∪
d
k=1{β∈Cd|βk∈πk(ZA∩Rd
≥0)}, whe e πkis he p ojec ion o he k- h
coo dina e. No ice also ha P⊆P.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17847
Rema k 4.16. In [7, Sec. 3] he au ho cons uc s ce ain Ge ey se ies solu ions ϕ k
o he hype geome ic sys em MB(γ ). Using Eule ’s e lec ion o mula, (z)(1−z)=
π/sin(πz) o z∈ Z, i can be easily shown ha , o all k∈and when βis gene ic
enough,
Sk(B−1
σγ;xσx−B−1
σBσ
σ)=πde√−1π|Aσk|
sin(π(−β+Aσk)) ·ϕ k.
The gene ici y condi ion he e means ha β/∈Pand ha β−Aσkdoes no ha e
in ege coo dina es o all k∈.
5 In eg als O e Rapid Decay Cycles
The goal o his sec ion is o p o e ha when β<0 is su icien ly gene al, he in eg als
s udied in Theo em 4.3, a e in ac in eg als o e apid decay cycles in he sense o
[15]. These in eg als a e de ined wi hou he condi ion β<0 and a e s ill solu ions
o ou GKZ sys em when βk≥0 o some k. By me omo phic con inua ion p o ed
in Theo em 4.11 hey admi asymp o ic expansions as Ge ey se ies solu ion o all β
su icien ly gene al in Cd.
5.1 Desc ip ion o apid decay cycles
In his sec ion we i s b ie ly ecall he heo y o apid decay homology by M. Hien in
[15, Sec. 5.1] and gi e a su icien condi ion o de ec a cycle o his homology.
Le Ube a complex quasi-p ojec i e a ie y o e Co dimension d.Le h∈O(U)
and le Xbe a smoo h p ojec i e compac i ica ion o U, such ha D=X Uis a no mal
c ossing di iso , and hex ends o a map h:X−→P1.
Le us deno e by π:
X(D)−→Xan he eal o ien ed blow-up along Das de ined
in [26, 8.2]. The space
X:=
X(D)can be embedded in o a eal Euclidian space as a semi-
analy ic subse , and hinduces a map
h:
X−→
P1, whe e
P1→P1is he eal blow-up o
in ini y.
Le us desc ibe he mo phism π, locally a p∈Dwi h local coo dina es 1,..., d
such ha p=0andD={ 1··· k=0},
π:([0, ) ×S1)k×B(0, )d−k−→ Cd
(( j,e√−1θj)k
j=1, )→ ( 1·e√−1θ1,..., k·e√−1θk, ),
whe e =( k+1,..., d)and >0 is a small eal numbe .
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17854 F.-J. Cas o-Jiménez e al.
(3) The p ojec ion o Sηon he space Rτ
>0is a bijec ion Sη−→Uη o he open
subse desc ibed by he inequali ies:
|a|< j∈ηaj
k
∈τ
a
o any k∈τ,
and he alue o he η-coo dina e ρo a poin ∈Sηis a unc ion ρ(( k)k∈τ)
by implici equa ion (5.6).
(4) Le s∈{0, ...,d}be he numbe o elemen s in η. Le us deno e Υ()
η he
union o he pieces o he cycle Υ() abo e he s a um Sη.ThenΥ()
η
is he union o 2d−spieces. A ypical piece is indexed by some (ξk)k∈τ∈
({0, 1})τ, and pa ame ized by k∈η[0, 2qkπ]×Uη⊂k∈η[0, 2qkπ]×Rτ
>0,in
he ollowing way:
(θj)j∈η;( k)k∈τ−→ (ρe√−1θj)j∈η;( ke2√−1πξkqk)k∈τ. (5.7)
(5) We choose he cohe en sys em o o ien a ions inspi ed by he p oduc o
cycles γk, wi h he ci cles posi i ely o ien ed: we o ien k∈η[0, 2qkπ]×Rτ
>0,
by i s canonical o ien a ion mul iplied by he signa u e o he pe mu a ion
(η,τ) o {1, ...,d},andby(−1)d−ξk.
In ac one can easily check ha he e is a adial iso opy om γ1×···×γd o Υ(),
which yields an o ien ed s a i ied isomo phism. Indeed, o :=( 1,..., d)∈(Rd
>0),
wi h k≥ o all k,de ine 0=min{ k}. On he hal -line R>0 he e is a unique poin
=(
1,...,
d)wi h min{
k}=, and a unique poin ρ:=(ρ1,...,ρd), such ha ρa=|a|
(see Figu e 3 o d=2). Le us conside ρ0=min{ρk}and he linea mul iplica ion
on R>0 by he a io ρ0/ = 0 −a/|a|, which depends con inuously on . Then he
map log +√−1θ→ log((ρ0/) )+√−1θ om γ1×···×γd o Υ() is he men ioned
adial iso opy.
P oposi ion 5.6. The wis ed cycle Υ() is a apid decay cycle. In pa icula , he
in eg als HΥ()(β;y)a e con e gen .
Rema k 5.7. Again we hink o Υ()as well as a cycle on (
C∗)d, o as a wis ed cycle
on ei he (C∗)d
uo (C∗)d
. I can be w i en as a sum u⊗ςuo ⊗ς , wi h e ms
in one- o-one co espondence by u= , and he b anches ςuand ς o uβa e
compa ible wi h he maps (
C∗)d−→(C∗)d
−→(C∗)d
uand yields he change o a iables
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17855
Fig. 3.
o mula deduced om (5.2):
u
u−βeM(u,y)du
u=
−q,βeM( q,y)
d
k=1
qk
d
.
Since he exponen M( q,y)is uni alen ou Co olla y 5.2 can be applied o Υ()
seen as a wis ed cycle on (C∗)d
endowed wi h he pullback C· −q,βo he local sys em
C·u−β. Howe e , all ou calcula ions can be done wi h he a iable u. Indeed bo h
a iables uand a e equi alen o he con ol o apid decay a in ini y, since u1:=
|uk|=| k|qk.
P oo o P oposi ion 5.6.Le us conside a s a um wi h η= ∅. The las monomial o
he a gumen o he exponen ial in F(β;y)sa is ies
|ynua(n)|=|yn||a(n)|. (5.8)
Fo η={1, ...,d} he ibe o e Sηis a compac subse o (
C∗)dand he in eg and
o HΥ()(β;y)is holomo phic o e i , so he e is no hing o p o e. Le us assume o
simplici y ha η={1, ...,s}wi h 1 ≤s<d. On he s a um Sηwe ha e 1=···= s=
ρ<. Thus, we imi a e he p oo o Lemma 3.2 ( ecall ha τA=σ∪{n}in ou case) o
ge an uppe bound o he eal pa o M(u,y)=−u1−···−ud+n
j=d+1yjua(j). Recall
ha by Assump ion 4.4,a(j)=d
k=1νjke(k) o j=d+1, ...,n−1 whe e νjk ≥0and
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17856 F.-J. Cas o-Jiménez e al.
|a(j)|=d
k=1νjk <1. Hence,
⎛
⎝
n−1
j=d+1
yj a(j)⎞
⎠=⎛
⎝
n−1
j=d+1
yj
d
k=1
νjke(k)⎞
⎠≤
n−1
j=d+1|yj|(d
max
k=1 e(k))d
k=1νjk
≤Kmax ⎛
⎝1, d
k=1
e(k)κ⎞
⎠≤Kmax(1, (s+ s+1+···+ d)κ),
(5.9)
whe e K=n−1
j=d+1|yj|and κ=maxn−1
j=d+1d
k=1νjk.Finally,using ha (−uj)≤ρ< o
j=1, ...,, (−uj)=− j o j=s+1, ...,dand (5.8), we ob ain
M(u,y)≤s− s+1−···− d+|yn||a|+Kmax(1, (s+ s+1+···+ d)κ). (5.10)
Since s+1+···+ d≤u1=ρs+ s+1+···+ d≤s+ s+1+···+ d o u∈Υ()
η,
we s ill ge an inequali y o ype (3.10). The e a e cons an s Cη,cη>0 (depending also
on ybu independen o ) such ha M(y,u)≤Cη−cηu1 o all u∈Υ()
η.
A close look a he a gumen ha p o es (5.10) shows ha we can w i e he
ollowing uppe bound o |M(u,y)|:
|M(u,y)|≤d+|yn||a(n)|+K(d+u1)κ.
This uppe bound, he ela ion (5.10) in he o m o inequali y M(y,u)≤Cη−cηu1
and he ac ha 0 <κ<1 p o e ha M(u,y)/M(u,y) ends o ze o as u1 ends o
in ini y. In pa icula he a gumen o M(u,y) ends o πalong Υ()
η.
We need a simila esul when η=∅.Weno ice ha M(u,y)is uni alen on he
2db anches o Υ()
∅. Following he p oo o inequali y (5.10) we ob ain he inequali y:
M(u,y)≤− 1−···− d−|yn| a(n)+Kmax(1, ( 1+···+ d)κ).
Since o he imagina y pa we ha e he inequali y |M(u,y)|≤|yn| a(n)+
K(u1)κ, we deduce ha i we se αn=|π−a g yn|∈[0, π/2[, we ha e o any δ>0and
any u1la ge enough a g M( ,y)∈]π−αn−δ,π+αn+δ].
Le us use a good compac i ica ion Xo (C∗)d, a eal blow-up π:
X−→Xo X
along D, and apply Co olla y 5.2. The beha iou o a g M( ,y)when u1→+∞and
he ac ha o any R>,Υ()∩{u|u1≤R}is compac imply ha Υ() is a apid
decay cycle.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17857
Le us p o e ha when β<0, he in eg al HΥ()(β;y) ends, when →0, o he
in eg al (5.4) mul iplied by he ac o
T:=
ξ∈{0,1}d
(−1)d−|ξ|exp 2√−1πβkqkξk=
d
k=1
(exp(2√−1πqkβk)−1)(5.11)
ha comes om he pa ame iza ion (5.7) o η=∅.Since(5.4) is clea ly he limi o he
piece o he in eg al HΥ()(β;y)o e Υ()
∅, i su ices o show ha he in eg als o e Sη
o η= ∅ end o ze o.
Le us assume again o simplici y ha η={1, ...,s}wi h 1 ≤s≤d. On each
piece o Υ()
η he pa ame e s a e
(θ1,...,θs, s+1,..., d)∈
k∈η
[0, 2qkπ]×Uη.
and he change o a iables om he pa ame iza ion (5.7) induces he ollowing esul s
in he di e en ac o s o he in eg and:
d
k=1
duk
uk=(√−1dθ1)∧···∧(√−1dθs)∧d s+1
s+1∧···∧d d
d
,
u−β=ρ−β1−···−βs −βs+1
s+1··· −βd
dexp ⎛
⎝√−1⎛
⎝−
s
j=1
βjθj−
d
k=s+1
2πβkqkξk⎞
⎠⎞
⎠,
|u−β|=ρ−(β1+···+βs)
d
=s+1
−β
exp ⎛
⎝
s
j=1βjθj+
d
k=s+1
2πβkqkξk⎞
⎠
≤−(β1+···+βs)
d
=s+1
−β
exp d
k=1
2π|βk|qk.
F om hese inequali ies and he ac ha he eal pa o he a gumen o he exponen ial
unc ion is bounded om abo e by
Cη−cη( s+1+···+ d)
wi h Cη,cη∈R>0independen o , o ∈[0, 0], we see ha he in eg al o e Υ()
η
ends o ze o when →0 as expec ed, because −(β1+···+βs)>0.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17858 F.-J. Cas o-Jiménez e al.
Finally, le us p o e ha he in eg al HΥ() does no depend on : ake 0 <
1<
2.
We conside Υ([1,2]), he noncompac (d+1)-cycle
∈[1,2]{}×Υ()
wi h o ien ed bounda y {1}×Υ(
1)−{2}×Υ(
2). Conside hen o R>
2 he compac
cycle ΥR=Υ([1,2])∩([1,2]×PR), whe e PRis he polydisk
PR={u∈Cd||u1|≤R,...,|ud|≤R}.
In eg als HΥ() a e o he o m HΥ() =Υ()ω, whe e ωis a holomo phic d– o m
independen o and hence i is a closed o m. We ha e
0=ΥR
dω=∂ΥR
ω.
The bounda y ∂ΥRis equal o
({1}×Υ(
1)) ∩([1,2]×PR)−({2}×Υ(
2)) ∩([1,2]×PR)+∂R.
Since by examining he pa ame iza ion (5.7) we see ha each d-dimensional piece o ∂R
is included in an hype plane uj=R, hence, he es ic ion o i o ωis ze o. We deduce
ha he in eg al o ωon Υ(
j)∩PR(which can eplace ({j}×Υ(
j))∩([1,2]×PR)because
ωdoes no depend on ) o j=1, 2 a e equal. Taking he limi when R−→∞ we ob ain
he esul
HΥ(
1)=HΥ(
2).
In he case o gene al p,δ, we keep he same cycle and wo k wi h he in eg al
HΥ()(β;y):=Υ()
u−β−1exp ⎛
⎝−
d
k=1
e√−1πδkuk+
n
j=d+1
zjua(j)⎞
⎠du,
whe e zj=e√−1π1+2p+δ,a(j)yjand he p oo is essen ially he same wi h only an easy
modi ica ion o inequali y (5.10).
In pa icula , he cycle
Dp,δin he s a emen o Theo em 5.3, is he image o Υ()
by k=uk·exp(√−1π1+2p+δ,a(k)).
Conclusion: The in eg al HΥ()(β;y)is analy ic as a unc ion o β∈Cd. Mo eo e ,
when β<0, e−√−1π1+2p+δ,βHΥ()(β;y)=T·Fp,δ(β;y), see (5.11). Hence, i equals he
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17859
me omo phic con inua ion T·
Fp,δ(β;y)ou side he union o hype planes Pdesc ibed in
Lemma 4.9.
When qkβk/∈Z o all k∈{1, ...,d}, he ac o Tis non ze o and we
ob ain a Ge ey se ies expansion o he in eg al along apid decay cycles HΥ()(β;y).
To check his las claim we ha e o ema k ha he se o poles o he analy ic
con inua ion
Fp,δ(β;y)is con ained in Pwhich is con ained in he se de ined by
T=d
k=1(exp (2√−1πqkβk)−1)=0. This la e se is, unde Assump ion 4.1, he se o
pa ame e s β=B−1
σγsuch ha γis called esonan o B(see [12, 2.9]).
Coming back o he gene al si ua ion o Theo em 4.3, he esul o his heo em
and he abo e conside a ions p o e he ollowing heo em:
Theo em 5.8. I Assump ion 4.1 is sa is ied and γ∈Cdis non esonan o B, hen
all he Ge ey solu ions o MB(γ ) along he hype plane xn=0 can be desc ibed as
linea combina ions o a ixed se o asymp o ic expansions o in eg al solu ions o
ype IC(γ ;x)along apid decay cycles.
Funding
This wo k was suppo ed by Minis e io de Ciencia, Inno ación y Uni e sidades MTM2016-75024-
P, FEDER, and FQM333-Jun a de Andalucía [ o F.-J.C.-J. and M.-C.F.-F.].
Acknowledgmen s
We would like o hank K. Takeuchi and S.-J. Ma suba a-Heo o hei sugges ions and use ul
commen s abou he con en o his a icle. We also hank wo anonymous e e ees o hei
ca e ul eading and help ul sugges ions. The 1s au ho would like o hank he Dépa emen
de Ma héma iques o he Uni e si y o Ange s (F ance) o i s suppo du ing he 1s s age o
his esea ch. The 3 d au ho would like o hank he Depa men o Algeb a and he Ins i u e
o Ma hema ics o he Uni e si y o Se ille (IMUS) o hei suppo and hospi ali y du ing he
p epa a ion o his pape .
Re e ences
[1] Adolphson, A. “Hype geome ic unc ions and ings gene a ed by monomials.” Duke Ma h.
J. 73, no. 2 (1994): 269–90.
[2] Aomo o, K. and M. Ki a. “Theo y o hype geome ic unc ions.” Sp inge Monog . Ma h.,
Sp inge -Ve lag, Tokyo, 2011.
[3] Assi, A., F.-J. Cas o-Jiménez, and M. G ange . “The G öbne an o an an-module.” J. Pu e
Appl. Algeb a 150, no. 1 (2000): 27–39.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
17860 F.-J. Cas o-Jiménez e al.
[4] Cas o-Jiménez, F.-J., M.-C. Fe nández-Fe nández, T. Koike, and N. Takayama. “I egula
modi ied A-hype geome ic sys ems.” T ans. Ame . Ma h. Soc. 367, no. 8 (2015): 5415–45.
[5] Cas o-Jiménez, F. J. and M. G ange . “Ge ey expansions o hype geome ic in eg als I.”
In . Ma h. Res. No . IMRN 2015, no. 5 (2015): 1338–70.
[6] Es e o , A. and K. Takeuchi. “Con luen A-hype geome ic unc ions and apid decay
homology cycles.” Ame . J. Ma h. 137, no. 2 (2015): 365–409.
[7] Fe nández-Fe nández, M. C. “I egula hype geome ic D-modules.” Ad . Ma h. 224, no. 5
(2010): 1735–64.
[8] Fe nández-Fe nández, M. C. and F. J. Cas o-Jiménez. “Ge ey solu ions o i egula hype -
geome ic sys ems in wo a iables.” J. Algeb a 339 (2011): 320–35.
[9] Fe nández-Fe nández, M. C. and F. J. Cas o-Jiménez. “Ge ey solu ions o he i egula
hype geome ic sys em associa ed wi h an a ine monomial cu e.” T ans. Ame . Ma h. Soc.
363, no. 2 (2011): 923–48.
[10] Gel and, I. M., M. I. G ae , and A. V. Zele insky. “Holonomic sys ems o equa ions and se ies
o hype geome ic ype.” Dokl. Akad. Nauk SSSR 295, no. 1 (1987): 14–9. T ansla ion in So ie
Ma h. Dokl. 36, no. 1 (1988): 5–10.
[11] Gel and, I.M., A. V. Zele insky, and M. M. Kap ano . “Hype geome ic unc ions and o ic
a ie ies (o hype geome ic unc ions and o al mani olds).” T ansla ed om Funk sional.
Anal. i P ilozhen. 23, no. 2 (1989): 12–26. T ansla ion in Func . Anal. Appl. 23, no. 2 (1989):
94–106. I. M. Gel and, A. V. Zele inskii˘and M. M. Kap ano , Co ec ion o he pape :
“Hype geome ic unc ions and o ic a ie ies.” [Funk sional. Anal. i P ilozhen. 23, no. 2
(1989): 12–26]; (Russian) Funk sional. Anal. i P ilozhen. 27, no. 4 (1993) 91; T ansla ion in
Func . Anal. Appl. 27, no. 4 (1993), 295 (1994).
[12] Gel and, I. M., M. M. Kap ano , and A. V. Zele insky. “Gene alized Eule in eg als and A-
hype geome ic unc ions.” Ad . Ma h. 84, no. 2 (1990): 255–71.
[13] Gel and, I. M. and M. I. G ae . “GG unc ions and hei ela ions o gene al hype geome ic
unc ions.” Le . Ma h. Phys. 50, no. 1 (1999): 1–28.
[14] Hien, M. “Pe iods o i egula singula connec ions on su aces.” Ma h. Ann. 337, no. 3
(2007): 631–69.
[15] Hien, M. “Pe iods o la algeb aic connec ions.” In en . Ma h. 178, no. 1 (2009): 1–22.
[16] Ho a, R. “Equi a ian D-modules.” a Xi :ma h /9805021 1 [ma h. RT].
[17] Lau en , Y. “Polygône de New on e b- onc ions pou les modules mic odi é en iels.” Ann.
Sci. Éc. No m. Supé . Q. Sé . 20, no. 3 (1987): 391–441.
[18] Lau en , Y. and Z. Mebkhou .. “Pen es algéb iques e pen es analy iques d’un d-module.”
Ann. Sci. École No m. Supé . (4) 32, no. 1 (1999): 39–69.
[19] Ma suba a-Heo, S.-J. “On Mellin–Ba nes in eg al ep esen a ions o GKZ hype geome ic
unc ions.” Kyushu Jou nal o Ma hema ics (2020), a Xi (1802) 04939 [ma h.CA].
[20] Ma suba a-Heo, S.-J. L. “Residue, and Eule in eg al ep esen a ions o GKZ hype geome ic
unc ions.” a xi (1801) 04075 2 [ma h.CA].
[21] Ma suba a-Heo, S.-J. “Eule and Laplace in eg al ep esen a ions o GKZ hype geome ic
unc ions.” a Xi (1904) 00565 [ma h.CA].
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022
Ge ey Expansions o Hype geome ic In eg als II 17861
[22] Ma use ich, L.F., E. Mille , and U. Wal he . “Homological me hods o hype geome ic
amilies.” J. Ame . Ma h. Soc. 18, no. 4 (2005): 919–41.
[23] Mebkhou , Z. “Le héo ème de posi i i é de l’i égula i é pou les D_X-modules.” The
G o hendieck Fes sch i , Vol. III, 83–132. P og . Ma h. 88. Bi khäuse Bos on, Bos on, MA,
1990.
[24] Oha a, K. and N. Takayama. “Holonomic ank o A-hype geome ic di e en ial-di e ence
equa ions.” J. Pu e Appl. Algeb a 213, no. 8 (2009): 1536–44.
[25] Pham, F. Singula i ies o In eg als. Homology, Hype unc ions and Mic olocal Analysis.
T ansla ed om he 2005 F ench o iginal, Uni e si ex , Sp inge . London: EDP Sciences, Les
Ulis, 2011.
[26] Sabbah, C. In oduc ion o S okes S uc u es.Lec u e No es in Ma h. 2060. Heidelbe g:
Sp inge , 2013 xi +249 pp.
[27] Sai o, M., B. S u m els, and N. Takayama. G öbne De o ma ions o Hype geome ic Di e -
en ial Equa ions. Algo i hms Compu . Ma h. 6. Be lin: Sp inge , 2000 iii+254 pp.
[28] Schulze, M. and U. Wal he . “I egula i y o hype geome ic sys ems ia slopes along
coo dina e subspaces.” Duke Ma h. J. 142, no. 3 (2008): 465–509.
[29] Takayama, N. “Modi ied A-hype geome ic sys ems.” Kyushu J. Ma h. 63, no. 1 (2009):
113–22.
Downloaded om h ps://academic.oup.com/im n/a icle/2021/23/17823/5648064 by Uni e sidad de Se illa use on 28 June 2022