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,...,∂ns 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].
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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
=1akx∂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.
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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].
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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
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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
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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
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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.
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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
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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
xσB−1
σe(k)
o k∈σ,
whe e s
xσis 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)
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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
nQp,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 .
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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()kyF(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.
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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+1ma()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).
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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√−1π2p+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(√−1π2p+1,−β)ρ−β−1exp(−ρ1−···−ρd)dρ.

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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 ywi 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)⎞
⎠ym
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).
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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 yin 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 yand 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 ya he o igin. This p o es he equali y (4.4)whenβ<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 ym
m!is equal o
A0
p⎛
⎝β−ma(n)−
n−1

j=d+1
mja(j)⎞
⎠=
e√−1π2p+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.
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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√−1π1+2p,−β+Aσq−β+Aσqyq
q!
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Ge ey Expansions o Hype geome ic In eg als II 17845
=e√−1π1+2p,−β
k∈
k+m∈k
e√−1π1+2p,Aσ(k+m)−β+Aσ(k+m)yk+m
(k+m)!
=e√−1π1+2p,−β
k∈
e√−1π1+2p,Aσk
k+m∈k
e√−1π1,Aσm
−β+Aσ(k+m)yk+m
(k+m)!
=e√−1π1+2p,−β
k∈
e√−1π2p,AσkSk(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√−1π2p+1,βFp,δ(β;y)
is (e√−1π2p,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√−1π2p,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
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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]whenβ<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.
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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 .
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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
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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,βeM( 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,βo 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 u1:=
|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
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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+1d
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≤u1=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−cηu1 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+u1)κ.
This uppe bound, he ela ion (5.10) in he o m o inequali y M(y,u)≤Cη−cηu1
and he ac ha 0 <κ<1 p o e ha M(u,y)/M(u,y) ends o ze o as u1 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(u1)κ, we deduce ha i we se αn=|π−a g yn|∈[0, π/2[, we ha e o any δ>0and
any u1la 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 u1→+∞and
he ac ha o any R>,Υ()∩{u|u1≤R}is compac imply ha Υ() is a apid
decay cycle. 
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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√−1πβkqkξk=
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=1βjθj+
d

k=s+1
2πβkqkξk⎞
⎠
≤−(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.
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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√−1π1+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(√−1π1+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−√−1π1+2p+δ,βHΥ()(β;y)=T·Fp,δ(β;y), see (5.11). Hence, i equals he
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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 .
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