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.
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Castro-Jiménez et al. (2021) “Gevrey Expansions of Hypergeometric Integrals II,” International Mathematics Research Notices, Vol. 2021, No. 23, pp. 17823–17861 Advance Access Publication January 7, 2020 https://doi.org/10.1093/imrn/rnz303 Gevrey Expansions of Hypergeometric Integrals II Francisco-Jesús Castro-Jiménez1, María-Cruz Fernández-Fernández1,∗and Michel Granger2 1Departamento de Álgebra e Instituto de Matemáticas-IMUS, Universidad de Sevilla, Av. Reina Mercedes s/n 41012 Sevilla, Spain and 2Université d’Angers, Département de Mathématiques, LAREMA, CNRS UMR n. 6093, 2 Bd. Lavoisier, 49045 Angers, France ∗Correspondence to be sent to: e-mail: [email protected] 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. 1 Introduction In [10] (see also [11,12]) the authors introduce and study A-hypergeometric systems and their solutions, generalizing many classical hypergeometric differential equations. General A-hypergeometric systems, also known as GKZ systems, are finitely generated D-modules, where D:=C[x]∂=C[x1,...,xn]∂1,...,∂nstands for the complex n-th Weyl algebra. Let us first recall some preliminary notions and results in D–module theory. Given a left D–ideal J⊆D, we consider the cyclic D–module M:=D/J. A solution fof Mis an element of a left D–module Fsuch that P·f=0, ∀P∈J. In this paper we only consider the cases when Fis either the space of holomorphic functions or the space Communicated by Prof. Masaki Kashiwara Received April 16, 2019; Revised October 04, 2019; Accepted October 10, 2019 © The Author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17824 F.-J. Castro-Jiménez et al. of Gevrey series (of order s∈R)alongY={xn=0}at p∈Y. We recall that such a Gevrey series is an expression of the form f=∞ m=0fmxm nwhere fm=fm(x1,...,xn−1) is holomorphic at pand ∞ m=0fmxm n/(m!)s−1is convergent at p. The smallest possible s (if any) so that this latter condition holds is called the Gevrey index of f. On the other hand, if u,v∈Rnsatisfy u+v∈Rn >0one can consider the graded ideal (or initial ideal)ofJwith respect to L=(u,v), denoted by inL(J), which is an ideal in the polynomial ring C[x,ξ]=C[x1,...,xn,ξ1,...,ξn], see, for example, [3, page 28]. Its zero set V(inL(J)) ⊆C2nis the L–characteristic variety of the cyclic D–module M=D/J, see, for example, [28, Definition 3.1]. If F=(u,v)with u=(0, ...,0),v=(1, ...,1)then Ch(M):=V(inF(J)) is simply called the characteristic variety of M.TheD–module M is said to be holonomic if the dimension of Ch(M)is n.Thesingular locus of Mis the Zariski closure of the image of Ch(M)\{ξ1= ··· = ξn=0}⊆C2nby the projection C2n−→ Cn,(x,ξ) → x. On the other hand, set V:=(−en,en), where en=(0, ...,0,1), and denote Ls:=F+(s−1)Vfor s>1. The Ls-characteristic variety is known to be locally constant with respect to s>1 except at a finite set of values called the slopes of Malong Y, see [17]. If Mis holonomic and it has a Gevrey solution with Gevrey index s>1alongYthen sisaslopeofMalong Y, see [18, Théorème 2.4.2] and [23] for a more general and stronger statement. The input data for a GKZ system is a pair (A,β) where βis a vector in Cdand A=(ak)=(a(1),...,a(n)) ∈(Zd)nisad×nmatrix whose -th column is a() and ZA:=d k=1Za(k)=Zd.Thetoric ideal IA⊂C[∂]:=C[∂1,...,∂n] is the ideal generated by the family of binomials ∂u−∂v, where u,v∈Nnand Au =Av (we assume 0 ∈N). Following [10,11], the hypergeometric ideal associated with the pair (A,β) is HA(β) :=DIA+D(E1−β1,...,Ed−βd), where Ek=n =1akx∂is the k−th Euler operator associated with the k-th row of A. The corresponding hypergeometric D-module (or A-hypergeometric system) is MA(β) := D HA(β) . In [11]and[1, Thm. 3.9] the authors prove that any hypergeometric system MA(β) is holonomic. Moreover, a characterization of the regularity of MA(β), in the sense of D–module theory [18,23], is provided in the series of papers [16,27,28]. The holonomic D-module MA(β) is regular if and only if the toric ideal IAis homogeneous for the standard grading in the polynomial ring C[∂]. In particular the condition to be regular for MA(β) is independent of the parameter vector β. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17825 The dimension of the space of germs of holomorphic solutions of MA(β) around a generic point in Cnequals d!Vol(A)if βis generic (see [11], [1, Cor. 5.20], and [22]). Here Ais the convex hull in Rdof the points 0,a(1),...,a(n), where 0∈Rdis the origin, and Vol(A)is its Euclidean volume. These holomorphic solutions are represented as –series in [11] (see also [24]and[7]) when βis generic enough. A. Adolphson considers in [1, Sec. 2] integral representations of solutions of MA(β) that involve exponentials of polynomial functions and appropriate integration cycles. In [6], A. Esterov and K. Takeuchi prove that the generic holomorphic solution spaces are in fact completely described by Adolphson’s integral representations along rapid decay cycles as introduced by M. Hien in [14]and[15]. Such type of integrals are also used in [20] and generalized in [21], where they are called Laplace integrals. The slopes, see [18], of MA(β) along coordinate subspaces are described in [28]. Their corresponding irregularity sheaves and Gevrey series solutions, defined in [23], are studied and described for generic parameters βin [7] (see also [8,9]). Moreover, in [4, Proposition 5.3 and Remark 5.4] these Gevrey series solutions of MA(β) are interpreted as asymptotic expansions of certain of its holomorphic solutions under some assumption on the Gevrey index of the series, via the so-called modified A-hypergeometric systems introduced in [29]. In [5], and when Ais a row matrix with positive integer entries, the authors develop a link between Gevrey series solutions of MA(β) and holomorphic solutions in sectors following Adolphson’s approach. They prove that any Gevrey series solution, along the singular support of the system MA(β), is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation. In this paper we further develop this link when the matrix A=(a(1),...,a(n)) ∈ (Zd)nsatisfies two conditions. Since the rank of Ais assumed to be d, we may also assume, after a possible reordering of the columns, that the 1st dcolumns of A determine a (d−1)-simplex σ. We further assume that Asatisfies the following two conditions (see Assumption 4.1): (1) the points a(d+1),...,a(n−1)belong to the interior of the convex hull σof σand the origin; and (2) the point a(n)is not in σand belongs to the open positive cone of σ.Figure1shows an example of an allowed column set configuration for a 2 ×5matrixA, where σis the triangle. Under these two conditions we have that Y={xn=0}is an irreducible component of the singular locus of MA(β) [1, Sec. 3], there is only one slope of MA(β) along Y[28] and, if βis generic enough, the dimension of the space of Gevrey series solutions of MA(β) along Yis d!Vol(σ)[7]. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17826 F.-J. Castro-Jiménez et al. Fig. 1. We prove in Theorem 4.3 that for generic β∈Cd, the space of Gevrey series solutions of MA(β), along the hyperplane Y, has a basis given by asymptotic expansions of holomorphic solutions of MA(β) described by Adolphon’s integral representations. These integrals are solutions of type IC(β;x)=IC(β;x1,...,xn):=C t−β−1exp n =1 xta()dt, where t=(t1,...,td),dt =dt1···dtdand Cruns over a finite set of cycles on the universal covering of (C∗)d. These are Borel–Moore cycles or cycles with closed support on the universal covering of (C∗)d, a notion for which we refer to [25, II,5.3]). Moreover, we prove in Theorem 5.8 that these cycles can be replaced by a set of rapid decay homology cycles in the sense of [15]. Here is a summary of the content of this paper. In Section 2we consider a general matrix Aas before but not necessarily satisfying previous conditions (1) and (2) (see Assumption 4.1). Following a construction in [13, Sec. 4.4], we describe cycles Cp,δin the universal covering of (C∗)d, depending on a given point x∈Cn.Wefixamaximal simplex σ⊂{1, ...,n}, that is, the set {a(k)|k∈σ}isabasisofRd. Then this cycle depends only on xσ:=(xk)k∈σ, and on vectors p∈Zσand δ∈Rσwith components δk satisfying |δk|<1/2. In Section 2.3 we give a sufficient condition for the integrand of Ip,δ(β;x):=ICp,δ(β;x)to have moderate growth along Cp,δ. This is a step towards sufficient conditions of convergence for Ip,δ(β;x)that are developed in Section 3. In Section 3, we perform the appropriate toric change of variables in the universal covering of (C∗)d, like in [13], which reduces the description of asymptotic Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17827 expansions for the integrals Ip,δ(β;x)to the study of integrals of type Fp,δ(β;y):=Dp,δ t−β−1exp ⎛ ⎝t1+···+td+ n j=d+1 yjta(j)⎞ ⎠dt, where the cycle Dp,δis the image of Cp,δunder the change of variables. The new integral Fp,δ(β;y)looks like a particular case of Ip,δ(β;x), with the 1st d×dsubmatrix (a(1),...,a(d)) equal to the identity matrix. However, the matrix A=(a(1),...,a(n)) is now allowed to have rational non integer coefficients. The crucial point for convergence statements is a condition of rapid decay at infinity, see inequality (3.10). We prove that, under some conditions, the integral Fp,δ(β;y)is absolutely convergent when βk<0for k∈σand y∈(C∗)n−d; see Lemmata 3.1 and 3.2. Section 4contains some of the main results of this paper. We assume that the matrix Adefined in Section 3satisfies more conditions in Assumption 4.4, deduced from conditions (1) and (2) in Assumption 4.1 already considered for the original matrix. First we prove that the conditions for convergence in Lemma 4.5, can be obtained in practice for every y∈Cn−dwith yn= 0. We fix p∈Zdand δ∈Rdonce for all and we omit these subindexes in our formulas. As a step towards previously mentioned Theorem 4.3, we prove in Theorem 4.7 that if β<0, there is an asymptotic expansion with respect to the variable ynin some sector in C∗: F(β;y)∼ yn→0 m∈N A(β;m,y)ym n m!, (1.1) where y=(yd+1,...,yn−1)and A(β;m,y):=Dp,δ t−β−1+ma(n)exp ⎛ ⎝t1+···+td+ n−1 j=d+1 yjta(j)⎞ ⎠dt. Assumption 4.4 plays an essential role in the proof of this result. Without assumption (1), we might need to impose further conditions on the arguments of y, e.g. conditions (3.5)jfor all j, in order to guarantee the convergence of F(β;y). Without condition (2), the vertex a(n)could have negative components and the integrals defining the coefficients A(β;m,y)would fail to be convergent for mlarge enough. Then we prove in Lemma 4.9 that F(β;y)admits a meromorphic continuation F(β;y), with respect to the variable β, with poles at most in a countable locally finite union of hyperplanes Pin Cd. The proof of this lemma uses that the points Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17828 F.-J. Castro-Jiménez et al. a(d+1),...,a(n)belong to d k=1R>0a(k)=Rd >0, which follows from conditions (1) and (2). The set Pis contained in the set of so-called resonant parameters of A[12,2.9]and it is explicitly described in terms of the columns of A.WealsoproveinLemma4.10 that, for any fixed parameter β∈ P, the meromorphic continuation F(β;y)admits an asymptotic expansion along yn=0 and that the coefficients A(β;m,y)of this expansion are the analytic continuation of the previously introduced A(β;m,y). In Section 5we prove that when β<0andβis sufficiently general, the integrals F(β;y)are in fact equal to integrals over rapid decay cycles in the sense of [15] (see Theorem 5.3). The statements involving Borel–Moore cycles are weaker because the analytic continuations are not expressed by integral along cycles when βk>0for some k. Another reason is that they are not cycles in the suitable homology adapted to the problem, like for Hien’s rapid decay homology. The notion of rapid decay cycles is explained in Section 5.1. Section 5.2 is devoted to the construction of rapid decay cycles. We start from a product of Hankel contours, along which the hypergeometric integrals are grossly divergent, but then we build a refined towards infinity version of this product along which convergent integrals are obtained. These integrals in Section 5are also defined when βk≥0 for some kand they are still solutions of MA(β). In Section 5.2 we prove, by using Section 4, that these integrals admit asymptotic expansions as Gevrey series solutions of MA(β) for non resonant βin Cd. 2 Products of Lines for Rapid Decay 2.1 Notations Let us slightly change our notation used in the introduction and let us start with a pair (B,γ), where B:=(b(1),...,b(n))∈(Zd)nisad×nmatrix, described as a list of columns such that ZB:=Zb(1)+···+Zb(n)=Zdand where γis a parameter vector in Cd.We are concerned with integrals: IC(γ ;x)=IC(γ ;x1,...,xn):=C t−γ−1exp n =1 xtb()dt, where 1=(1, ...,1)∈Ndand Cis a suitable cycle. To make precise this definition let us specify some conventions and notations. As already mentioned, Cis a cycle on the universal covering ( C∗)dof (C∗)d. We identify ( C∗)dwith Cdor with Rd >0×Rdand write z=(log r+√−1θ) or (r,θ), respectively, for the coordinates on ( C∗)dwith θka branch of arg tktk=exp(zk),andrk=|tk|. We set, for Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17829 any vector v∈Cd,tv=d k=1tvk k. This is a multivalued monomial, namely the function on the universal covering: exp z,v=exp d k=1 vk(log rk+√−1θk), where we set, given two vectors u,v∈Cd,u,v=d k=1ukvk. 2.2 Description of cycles of rapid decay at infinity If τ⊂{1, ...,n}, we denote by Bτthe matrix whose columns are b(j)with j∈τand by τ the complement of τin {1, ...,n}. Recall that a subset σ⊂{1, ...,n}is called a maximal simplex for Bif the columns {b(k),k∈σ}form a basis of Rd. Such a maximal simplex σis also called a base in [11, Sec. 1.1]. We often identify the set σwith the set of columns {b(k),k∈σ}. We fix a maximal simplex σfor Band take x∈Cnsuch that xk= 0 for all k∈σ. We also fix p=(pk)k∈σ∈Zσ≃Zd,δ=(δk)k∈σ∈Rσ≃Rdsuch that |δk|<1 2for all k∈σ. We denote by Cp,δthe cycle in the space ( C∗)ddescribed by the following condition on the argument θ:=arg tof t∈(C∗)d(i.e. tθ:=(arg t1,..., arg td)): arg(xktb(k))=arg xk+b(k),θ=(1+δk+2pk)π for allk∈σ. (2.1) Remark 2.1. The cycle Cp,δdepends on xσ:=(xk)k∈σ∈(C∗)σ≃(C∗)dandalsoona choice of its argument. However, a change in this choice yields only a reindexation by p of the unchanged set of these cycles. For that reason in all our statements we stick on xσ∈(C∗)σwithout passing to the universal covering of (C∗)σ. From now on we will denote Ip,δ(γ ;x)=ICp,δ(γ ;x). The cycles Cp,δareaslightly modified version of cycles considered in [13, Sec. 4.4]. Let us set :=π 2,3π 2+2πZ. The equality (2.1) can be globally rewritten using matrix notation: arg xσ+tBσθ=(1+δ+2p)π ∈σ. (2.2) There is a unique solution θof the previous equation θ=(tBσ)−1−arg xσ+(1+δ+2p)π(2.3) so that Cp,δis the cartesian product of dopen half–lines. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17830 F.-J. Castro-Jiménez et al. Given p,p∈Zd,letθ=arg t,θ=arg tbe the corresponding unique solutions for equation (2.2). If (tBσ)−1(p−p)∈Zdthen θ−θ∈2πZdand the projections of the two cycles Cp,δ and Cp,δon (C∗)dare the same. We check that the convergence of the two integrals along the cycles Cp,δand Cp,δare then equivalent to each other and, moreover, the integral solutions differ only by a constant factor: Ip,δ(γ ;x)=Cp,δ t−γ−1exp n =1 xtb()dt=e−2π√−1(d k=1mkγk)Ip,δ(γ ;x) for some mk∈Z,k=1, ...,d. When pvaries in a set of representatives of Zd ZtBσ, we will see that the convergence of the integral Ip,δ(γ ;x)depends on δ(see Remark 2.2 and Lemma 4.5). However, choosing in each such class an appropriate δ, we can find, as a consequence of our main result and under some conditions (see Assumption 4.1), [Zd:ZtBσ]=|det Bσ| many integral solutions Ip,δ(γ ;x)which are linearly independent (see Theorem 4.3). We will see in the proof of Lemma 3.2, after the change of variables defined in Section 3, that the cycles Cp,δare of rapid decay at infinity. 2.3 Sufficient conditions for moderate growth Sufficient conditions for the convergence of the integral Ip,δ(γ ;x)aredetailedinthe next section (see Lemma 3.2 and Remark 3.6). As a preliminary step let us look here at a condition for bounding the exponential term in that integral; let us notice that condition (2.1) implies that (xktb(k))<0alongCp,δfor any xk∈C∗,k∈σ. If we additionally could ensure that arg(xjtb(j))∈for all j∈σsuch that xj= 0 (2.4) (see Remark 2.2 below) then the argument of the exponential has negative real part along Cp,δ; hence, the absolute value of the exponential term in the integral Ip,δ(γ ;x)is bounded by 1. Then if we take into account the term t−γ−1, the integrand of Ip,δ(γ ;x)has moderate growth along Cp,δ. Remark 2.2. Let us notice that condition (2.3) determines a unique cycle Cp,δfor a given pand δ. It is not clear that for given x∈(C∗)σ×Cσand p∈Zσone can always choose δ∈Rσfor this cycle to satisfy conditions (2.2)and(2.4). It is therefore interesting to weaken these conditions by keeping only the significant ones. In Lemma 3.2 and Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17831 Remark 3.6, completed by Remark 3.3, we do this with a reduced version of the variables xrenamed y. We will see in the next section that conditions (2.2)and(2.4) are sufficient convergence conditions for the integrals Ip,δ(γ ;x)when combined with a condition on the parameter γ. After an appropriate change of variables we can interpret them as a condition of rapid decay at infinity, see the proof of Lemma 3.2. We notice that Cp,δis a Borel–Moore cycle in ( C∗)dbut not in general a rapid decay cycle in the sense of [14], see Remark 3.5. However, we shall prove in Section 5that the integral along Cp,δis equal to an integral along a rapid decay cycle (see Theorem 5.3) under Assumption 4.4,andfor values of γthat guarantee convergence. Our result can be then interpreted in the frame of [6, Th. 4.5]. 3 A Change of Variables and Explicit Calculations We will assume for simplicity, after a possible reordering of the variables, that the maximal simplex σis {1, ...,d}.Letusfixx∈(C∗)d×Cn−d, and an argument of all xk with k∈σ. We consider the finite to one covering (C∗)d→(C∗)dof degree det Bσ,given by the formula: sk=xktb(k)for k∈σ. We think of it as a (ramified) toric change of variables. We fix a branch of log xσand we consider the bijective change of variables on the universal covering ( C∗)d≃Cd,givenby log sk−log xk=log t·b(k). Fractional powers like xv σwith v∈Qdhave the natural meaning xv σ=exp(log xσ· v), and the inverse mapping on ( C∗)dcan be read as follows using these fractional powers: tk=s xσB−1 σe(k) for k∈σ, where s xσis the vector with coordinates sk/xkand (e(k))k∈σis the standard basis of Zd. The image Dp,δof the cycle Cp,δdescribed in Section 2, is determined by the conditions: arg sk=(1+δk+2pk)π for all k∈σ. (3.1) Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17838 F.-J. Castro-Jiménez et al. Remark 4.6. Let us fix p,δand a branch αnof arg yn,0 satisfying condition (??)n.This defines the subset R⊂Rof allowed arguments arg yn.LetSp,δ⊂C∗be the sector defined by arg yn∈R0the connected component of αnin R. Like in Remark 3.3,Sp,δis independent of the choice of αn. Theorem 4.7. If βk<0 for all k=1, ...,d, then for any given yn,0 ∈C∗there is an asymptotic expansion with respect to the variable ynin the open sector Sp,δ: Fp,δ(β;y)∼ yn→0 m∈N Ap,δ(β;m,y)ym n m!, where y=(yd+1,...,yn−1)and Ap,δ(β;m,y):=Dp,δ t−β−1+ma(n)exp ⎛ ⎝t1+···+td+ n−1 j=d+1 yjta(j)⎞ ⎠dt. Proof. We have to prove that for any integer N>0 there exists KN=KN(β,y)>0 such that Fp,δ(β;y)− N−1 m=0 Ap,δ(β;m,y)ym n m!≤KN|yn|N holds for every yn∈Sp,δ. Let N(z):=ez− N−1 m=0 zm m! for z∈C.Thenwehave |N(z)|≤|z|N N!for all zsuch that z<0. Recall that by the assumption on δwe have (ynta(n))<0whent∈Dp,δsince yn∈Sp,δ.Thus,wehave Fp,δ(β;y)− N−1 m=0 Ap,δ(β;m,y)ym n m!=yN nQp,δ(β;y,N), where Qp,δ(β;y,N)=Dp,δ ta(n)N−β−1exp ⎛ ⎝t1+···+td+ n−1 j=d+1 yjta(j)⎞ ⎠N(ynta(n)) (ynta(n))Ndt. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17839 The absolute value of the integrand in Qp,δ(β;y,N)is bounded by the function 1 N!ta(n)N−β−1exp ⎛ ⎝t1+···+td+ n−1 j=d+1 yjta(j)⎞ ⎠, which is independent of ynand integrable over Dp,δby Lemma 3.2 (that can be applied to the submatrix of Adefined by its 1st n−1 columns because of Assumption 4.4). Notice that we use here that (β −a(n)N)<0 for all N>0sincea(n)does not have negative coordinates. Thus, there exists KN=KN(β,y)>0 such that Qp,δ(β;y,N)≤KN. This finishes the proof. Remark 4.8. Notice that Ap,δ(β;m,y)=Fp,δ(β −ma(n);y)for the submatrix of A defined by its 1st n−1 columns. In particular it is analytic with respect to (β,y)by Remark 3.4. We extend Theorem 4.7 to nonnegative values of βkin Section 4.2. 4.2 Analytic continuation with respect to β In this section we focus on the analytic dependency of F(β;y)=Fp,δ(β;y)on β.Letus take yn,0 ∈C∗and p∈Zd. We choose δas in Lemma 4.5 and we omit p,δin the remainder of this subsection. We assume now that ybelongs to Cn−d−1×Sp,δ, where the sector Sp,δ is defined in Remark 4.6. The integral F(β;y)is a solution of the reduced GG-system (see [13]): βkF(β;y)= n =d+1 a()kyF(β −a();y)+F(β −e(k);y)fork=1, ...,d(4.1) F(β −a();y)=∂F ∂y (β;y)for=d+1, ...,n. (4.2) Lemma 4.9. The function F(β;y)admits a meromorphic continuation with respect to β, denoted by F(β;y), with poles at most along the countable locally finite union of hyperplanes P:= d k=1{β∈Cd|βk∈πk(NA)}, where NA=Na(1)+···+Na(n)and πk:Qd→Qdenotes the projection to the k-th coordinate. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17840 F.-J. Castro-Jiménez et al. Proof. The initial domain of analyticity of F(β;y)is defined by βk<0 for all k=1, ...,d. Let us fix conditions βk<0fork=2, ...,dand extend the domain of analyticity in the coordinate β1using equation (4.1)1as follows. The functions F(β −a();y)for =d+1, ...,nand F(β −e(1);y)are analytic for β1< a1:= min{a()1,1}and hence it follows from equation (4.1)1that F(β;y)is meromorphic in β1< a1with at most a pole in β1=0. In the general inductive step for the variable β1, we assume that F(β;y)is meromorphic in the half-space β1<(q−1) a1. Then, on the domain defined by β1<q a1, the right-hand side of (4.1)1is meromorphic, with poles of type β1=c+1, or β1=c+a()1, where β1=cruns over all the poles of F(β;y). We obtain that F(β;y) is also meromorphic in the same domain adding these new poles to those already found. Thus, by induction, we get that F(β;y)is also meromorphic for β1∈Cand βk<0fork=2, ...,dwith poles at most along β1=n =d+1ma()1+m, for all md+1,...,mn,m∈N. By an analogous argument in k=2, ...,dwe get the result. Notice that the equations (4.2) are then satisfied by F(β;y)by analytic continuation on U:=Cd\P. Lemma 4.10. For any fixed β∈, F(β;y)admits an asymptotic expansion along yn= 0inSp,δ. Furthermore, the coefficients A(β;m,y)of this expansion are analytic with respect to β∈U. Hence, they are analytic continuations of the coefficients A(β;m,y) described in Theorem 4.7. Proof. It follows from an induction starting from Theorem 4.7 and parallel to the one used in the proof of Lemma 4.9 that for any fixed β∈, F(β;y)admits asymptotic expansions along yn=0inSp,δ. By construction, these analytic continuations satisfy equation (4.1), for any β∈U. This implies that the coefficients A(β;m,y)of these expansions satisfy the following equations for 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 induction like in Lemma 4.9,using(4.3) and Remark 4.8, A(β;m,y) is analytic with respect to βand y, hence as a function of βit is an analytic continuation to Uof A(β;m,y). Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17841 We have proved the following theorem that implies the last sentence in Theorem 4.3 when we return to the integrals IC(β;y). Theorem 4.11. There is an asymptotic expansion along yn=0, in an appropriate open sector Sp,δaround any half-line R>0·yn,0 ⊂C∗: Fp,δ(β;y)∼ yn→0 m∈N Ap,δ(β;m,y)ym n m!, where Ap,δ(β;m,y)is the analytic continuation of Ap,δ(β;m,y)to β∈U. 4.3 Parametrizations We go on working with the reduced form of the integral described in (3.2–3.4), and we study integrals of the form Fp,δ(β;y)=Dp,δ t−β−1exp ⎛ ⎝t1+···+td+ n j=d+1 yjta(j)⎞ ⎠dt. Lemma 4.12. If β<0, then A0 p,δ(β) :=Dp,δ t−β−1exp(t1+···+td)dt=e√−1π2p+1,−β(−β), where (−β) :=d k=1(−βk). Proof. The integrand t−β−1exp(t1+···+td)dtis of rapid decay at infinity in the product of dsectors defined by the condition: arg(tk)∈[(1+min{0, δk}+2pk)π,(1+max{0, δk}+2pk)π], k∈σ. Thus, since this product of sectors contain Dp,0 and Dp,δ, we know by elementary considerations in one complex variable, that A0 p,δ(β) does not depend on δk∈]−1 2,1 2[and so A0 p,0(β) =A0 p,δ(β). We parametrize Dp,0 by tk=ρke√−1π(2pk+1)=−ρkwith ρk∈]0, +∞),andthe result follows directly from the expression that we obtain A0 p,0(β) =]0,+∞)dexp(√−1π2p+1,−β)ρ−β−1exp(−ρ1−···−ρd)dρ. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17842 F.-J. Castro-Jiménez et al. Since A0 p,δ(β) does not depend on δ, from now on we drop δand set A0 p(β) := A0 p,0(β) =A0 p,δ(β). We notice that Fp,δ(β;y)is locally constant with respect to δby a similar homotopy argument. However, the dependency on δof Fp,δ(β;y)must be kept because the argument by homotopy works only for small perturbations of δ. This does not allow a reduction of δto zero. Let us now make the analytic continuation of the coefficients of the asymptotic expansion described in Theorem 4.7 more precise by developing them with respect to y. Lemma 4.13. The coefficients of the asymptotic expansion described in Theorem 4.7 are analytic functions of the variables ywith the following power series development: 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) Furthermore, this expansion is still valid for the meromorphic continuation of Ap,δ(β;m,y)found in Lemma 4.10 and the meromorphic continuation of A0 p(β) deduced from Lemma 4.12. Proof. Recall that, when β<0 the coefficient we consider has the form Ap,δ(β;m,y)=Dp,δ ϕ(β;y;t)dt with ϕ(β;y;t)=t−β−1+ma(n)exp ⎛ ⎝t1+···+td+ n−1 j=d+1 yjta(j)⎞ ⎠. We set |tk|=ρkfor k=1, ...,dand we parametrize Dp,δby ρ∈Rd >0.Wefixa polydisc Q={y||yj|<Rj,j=d+1, ...,n−1}⊂Cn−1−d. The function ϕ(β;y;t)is holomorphic with respect to y∈Cn−d−1. By the same argument as in the proof of Lemma 3.2 and inequality (3.10), the integrand ϕ(β;y;t)dt is dominated, via the parametrization tk=e(1+δk+2pk)√−1πρkand up to a constant factor, by ρ−β+ma(n)−1exp(C−c(ρ1+···+ρd))dρ for some constants C,c∈R>0. These constants depend only on Qbut not on y∈Qby Remark 3.4 applied to Ap,δ(β;m,y)instead of Fp,δ(β;y). Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17843 For each j=d+1, ...,n−1, the integral Dp,δ ∂ϕ(β;y;t) ∂yj dt has an expression similar to the one for Ap,δ(β;m,y),withβreplaced by β−a(j).Bythe same argument as for ϕ, the integrand ∂ϕ(β;y;t) ∂yjdtis dominated, up to a constant factor, by ρ−β+ma(n)+a(j)−1exp(Cj−cj(ρ1+···+ρd))dρ for some constants Cj,cj∈R>0, independent of yin the polydisk Q. By Lebesgue’s theorem on dominated convergence for integrals, these considerations prove that Ap,δ(β;m,y)is holomorphic with respect to yand that ∂Ap,δ(β;m,y) ∂yj=Dp,δ ∂ϕ(β;y;t) ∂yj dt for all j=d+1, ...,n−1. If we iterate the argument we obtain an expression of the partial derivatives of Ap,δ, up to any order m=(md+1,...,mn−1): ∂|m|Ap,δ(β;m,y) ∂md+1yd+1···∂mn−1yn−1=Dp,δ ∂|m|ϕ(β;y;t) ∂md+1yd+1···∂mn−1yn−1 dt. Setting y=0 in this last expression gives the coefficients of the Taylor expansion of Ap,δ(β;m,y)with respect to yat the origin. This proves the equality (4.4)whenβ<0. The last claim of this lemma follows from the explicit calculation in Lemma 4.12 from which we see that the coefficient of ym m!is equal to 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 the standard properties of the -function, this coefficient admits a meromorphic continuation with respect to β, with poles along a subset of Pdefined in Lemma 4.9. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17844 F.-J. Castro-Jiménez et al. When β∈Cd\Pthe right-hand side of (4.4) is still defined and yields a convergent power series defined for all y∈Cn−d−1because of the conditions |a(j)|<1 for j=d+1, ...,n−1, and standard estimates on -functions (see, e.g., [7, Lemma 3.8]). Therefore, it is an analytic continuation of the power series defined for β<0. Theequality(4.4) follows everywhere in Cd\Pwith the meromorphic continuation of Ap,δ(β;m,y)on the left-hand side, defined in Lemma 4.10. Remark 4.14. Notice that as a consequence of Lemma 4.13 the function Ap,δ(β;m,y) does not depend on δ. 4.4 Space of asymptotic expansions and Gevrey series In this section we finish the proof of Theorem 4.3. For any k∈Nn−d, let us set k:={k+m=(kd+1+md+1,...,kn+mn)∈Nn−d:Aσm∈Zd} and define Sk(β;y):= k+m∈k e|Aσ(k+m)|π√−1−β+Aσ(k+m))yk+m (k+m)!. Notice that the coefficients of the series Skare meromorphic with respect to β∈Cdwith at most simple poles along each hyperplane in P. In particular, if β/∈Pall these series are well-defined nonzero power series with support equal to ksince the Gamma function does not have any zero. It can be proved by using standard estimates of Gamma functions that these series are Gevrey along yn=0 with Gevrey index |a(n)|>1. Let ⊆Nn−dbe a set of cardinality [ZA:ZAσ] such that {Aσk+ZAσ:k∈}=ZA/ZAσ=ZA/Zd. We notice that the existence of such ⊆Nn−dfollows from [7, Lemma 3.2]. It is clear that G={Sk(β;y):k∈}is a linearly independent set because the series Sk have pairwise disjoint supports k. Using Theorem 4.7, Lemma 4.12, and Lemma 4.13, we have 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 from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals 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). Notice that previous power series is formal with respect to yn, with convergent coefficients. More precisely, it is a Gevrey series along yn=0 with Gevrey index |a(n)|>1. We notice also that β<0 implies that (β −Aσq)<0 for all q∈Nn−d,by using Assumption (4.4), which guarantees the convergence of all the integrals involved in Section 4.3. By the last claim in Lemma 4.13, this calculation is valid everywhere in the domain of analytic continuation Cd\P, since the argument applies also to the coefficients of the series Sk(β;y). The matrix of coefficients of the series Sk(β;y)in the asymptotic expansions of the functions e√−1π2p+1,βFp,δ(β;y) is (e√−1π2p,Aσk)k,p, where kvaries in .Ifpvaries in an appropriate set of [ZA:Zd] elements, this matrix is square and invertible. Indeed, we have ZA/ZAσ=ZA/Zd≃ Zd/ZM, where Mis the matrix of coordinates of the canonical basis of Zdwith respect to a basis of ZA/Zd. Thus, the matrix (e√−1π2p,Aσk)k,pis invertible by [19, Proposition 6.3], if pruns in a set of representatives of the quotient Zd/ZtM. In particular, if β/∈P the set of holomorphic functions Fp,δ(β;y), where pvaries in this set of representatives, is a linearly independent set and any Gevrey series along yn=0 in the space generated by the series {Sk(β;y):k∈}is an asymptotic expansion of a linear combination of the integrals Fp,δ(β;y). Now if we start from the matrix Bin Section 2and we apply the above results with the matrix A=B−1 σB=(I,B−1 σBσ)and the parameter β=B−1 σγ, we obtain a similar statement for the integrals IC(γ ;x)using (3.3)and(3.4) if we set yj=xjx−a(j) σfor all j=d+1, ...,n,ory=xσx−B−1 σBσ σ. Precisely, Mcan be chosen to be Bσ. We get that xB−1 σγ σ·Gis a linearly independent set of Gevrey series solutions of MB(γ ) along xn=0 with Gevrey index |a(n)|=|B−1 σb(n)|>1ifβ/∈P. Again this transformation involves a Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17846 F.-J. Castro-Jiménez et al. choice of arg xσ, but by Remark 2.1 and (3.3), a change in this choice does not modify the basis Gexcept for constant factors. It is enough to prove that the dimension of the space of Gevrey series solutions of MB(γ ) along xn=0isatmostequalto||=[Zd:ZBσ]whenβ<0. To this end, notice first that, if f=∞ m=0 fm(x1,...,xn−1)xm n is a Gevrey series belonging to this space, then the initial part of fwith respect to the weight vector w=(0, ...,0,1)∈Rnhas the form inw(f)=fm(x1,...,xn−1)xm nfor some m≥0 and it is hence a holomorphic function. Thus, by the same argument as in the proof of [27, Th. 2.5.5], it is a (holomorphic) solution of in(−w,w)(HB(γ )).This last ideal is the initial ideal with respect to wof the hypergeometric ideal associated with (B,γ) (see [27, p. 4]). In particular, the dimension of the space of Gevrey solutions is at most equal to the rank of in(−w,w)(HB(γ )), because one can choose a basis of Gevrey solutions of MB(γ ) such that their initial parts are also linearly independent (see [27, Proposition 2.5.7]). On the other hand, by using [27, Lemma 2.1.6] for (u,v)=(0,1)and (u,v)= (−w,w), we have that the initial ideal of in(−w,w)(HB(γ )) with respect to (0,1)is in(0,1)(in(−w,w)(HB(γ ))) =inL(HB(γ )) for L=(−w,w)+(0,1)with >0 small enough. Thus, by [28, Th. 4.21, Rk. 4.23, and Th. 4.28] for L=(−w,w)+(0,1)and Assumption 4.1, we have that the holonomic rank of in(−w,w)(HB(γ )) equals ||if γis not rank–jumping for B(i.e., if rank(MB(γ )) =d!Vol(B)), a condition that is weaker than β=(B−1 σγ) < 0by[1, Th. 5.15] (see also [27, Cor. 4.5.3]). This finishes the proof of Theorem 4.3. Remark 4.15. Notice that the proof of Theorem 4.3 shows that the constructed set of Gevrey series solutions xB−1 σγ σ·Gis still a basis of the space of Gevrey solutions of MB(γ ) along xn=0whenγis not rank-jumping and β=B−1 σγ/∈P, where Pis defined in Lemma 4.9. We do not know if under Assumption 4.1 the condition of γbeing rankjumping implies β∈P. However, it is true that if γis rank-jumping then it is semiresonant [1]. In particular, under Assumption 4.1,γis semi-resonant for Bif and only if β∈P:=∪ d k=1{β∈Cd|βk∈πk(ZA∩Rd ≥0)}, where πkis the projection to the k-th coordinate. Notice also that P⊆P. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17847 Remark 4.16. In [7, Sec. 3] the author constructs certain Gevrey series solutions ϕvk for the hypergeometric system MB(γ ). Using Euler’s reflection formula, (z)(1−z)= π/sin(πz)for z∈ Z, it can be easily shown that, for all k∈and when βis generic enough, Sk(B−1 σγ;xσx−B−1 σBσ σ)=πde√−1π|Aσk| sin(π(−β+Aσk)) ·ϕvk. The genericity condition here means that β/∈Pand that β−Aσkdoes not have integer coordinates for all k∈. 5 Integrals Over Rapid Decay Cycles The goal of this section is to prove that when β<0 is sufficiently general, the integrals studied in Theorem 4.3, are in fact integrals over rapid decay cycles in the sense of [15]. These integrals are defined without the condition β<0 and are still solutions of our GKZ system when βk≥0 for some k. By meromorphic continuation proved in Theorem 4.11 they admit asymptotic expansions as Gevrey series solution for all β sufficiently general in Cd. 5.1 Description of rapid decay cycles In this section we first briefly recall the theory of rapid decay homology by M. Hien in [15, Sec. 5.1] and give a sufficient condition to detect a cycle for this homology. Let Ube a complex quasi-projective variety over Cof dimension d.Leth∈O(U) and let Xbe a smooth projective compactification of U, such that D=X\Uis a normal crossing divisor, and hextends to a map h:X−→P1. Let us denote by π: X(D)−→Xan the real oriented blow-up along Das defined in [26, 8.2]. The space X:= X(D)can be embedded into a real Euclidian space as a semianalytic subset, and hinduces a map h: X−→ P1, where P1→P1is the real blow-up of infinity. Let us describe the morphism π, locally at p∈Dwith local coordinates t1,...,td such that p=0andD={t1···tk=0}, π:([0, ) ×S1)k×B(0, )d−k−→ Cd ((rj,e√−1θj)k j=1,t)→ (r1·e√−1θ1,...,rk·e√−1θk,t), where t=(tk+1,...,td)and >0 is a small real number. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17854 F.-J. Castro-Jiménez et al. (3) The projection of Sηon the space Rτ >0is a bijection Sη−→Uηto the open subset described by the inequalities: |a|<rj∈ηaj k ∈τ ra for any k∈τ, and the value of the η-coordinate ρof a point r∈Sηis a function ρ((rk)k∈τ) by implicit equation (5.6). (4) Let s∈{0, ...,d}be the number of elements in η. Let us denote Υ() ηthe union of the pieces of the cycle Υ() above the stratum Sη.ThenΥ() η is the union of 2d−spieces. A typical piece is indexed by some (ξk)k∈τ∈ ({0, 1})τ, and parametrized by k∈η[0, 2qkπ]×Uη⊂k∈η[0, 2qkπ]×Rτ >0,in the following way: (θj)j∈η;(rk)k∈τ−→ (ρe√−1θj)j∈η;(rke2√−1πξkqk)k∈τ. (5.7) (5) We choose the coherent system of orientations inspired by the product of cycles γk, with the circles positively oriented: we orient k∈η[0, 2qkπ]×Rτ >0, by its canonical orientation multiplied by the signature of the permutation (η,τ) of {1, ...,d},andby(−1)d−ξk. In fact one can easily check that there is a radial isotopy from γ1×···×γdto Υ(), which yields an oriented stratified isomorphism. Indeed, for r:=(r1,...,rd)∈(Rd >0), with rk≥for all k,definer0=min{rk}. On the half-line R>0rthere is a unique point r=(r 1,...,r d)with min{r k}=, and a unique point ρ:=(ρ1,...,ρd), such that ρa=|a| (see Figure 3for d=2). Let us consider ρ0=min{ρk}and the linear multiplication on R>0rby the ratio ρ0/ =r0r−a/|a|, which depends continuously on r. Then the map log r+√−1θ→ log((ρ0/)r)+√−1θfrom γ1×···×γdto Υ() is the mentioned radial isotopy. Proposition 5.6. The twisted cycle Υ() is a rapid decay cycle. In particular, the integrals HΥ()(β;y)are convergent. Remark 5.7. Again we think of Υ()as well as a cycle on ( C∗)d, or as a twisted cycle on either (C∗)d uor (C∗)d v. It can be written as a sum u⊗ςuor v⊗ςv, with terms in one-to-one correspondence by u=v, and the branches ςuand ςvof uβare compatible with the maps ( C∗)d−→(C∗)d v−→(C∗)d uand yields the change of variables Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17855 Fig. 3. formula deduced from (5.2): u u−βeM(u,y)du u=v v−q,βeM(vq,y) d k=1 qk dv v. Since the exponent M(vq,y)is univalent our Corollary 5.2 can be applied to Υ() seen as a twisted cycle on (C∗)d vendowed with the pullback C·v−q,βof the local system C·u−β. However, all our calculations can be done with the variable u. Indeed both variables uand vare equivalent for the control of rapid decay at infinity, since u1:= |uk|=|vk|qk. Proof of Proposition 5.6.Let us consider a stratum with η= ∅. The last monomial of the argument of the exponential in F(β;y)satisfies |ynua(n)|=|yn||a(n)|. (5.8) For η={1, ...,d}the fiber over Sηis a compact subset of ( C∗)dand the integrand of HΥ()(β;y)is holomorphic over it, so there is nothing to prove. Let us assume for simplicity that η={1, ...,s}with 1 ≤s<d. On the stratum Sηwe have r1=···=rs= ρ<. Thus, we imitate the proof of Lemma 3.2 (recall that τA=σ∪{n}in our case) to get an upper bound for the real part of M(u,y)=−u1−···−ud+n j=d+1yjua(j). Recall that by Assumption 4.4,a(j)=d k=1νjke(k)for j=d+1, ...,n−1 where νjk ≥0and Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17856 F.-J. Castro-Jiménez et al. |a(j)|=d k=1νjk <1. Hence, ⎛ ⎝ n−1 j=d+1 yjra(j)⎞ ⎠=⎛ ⎝ n−1 j=d+1 yj d k=1 rνjke(k)⎞ ⎠≤ n−1 j=d+1|yj|(d max k=1re(k))d k=1νjk ≤Kmax ⎛ ⎝1, d k=1 re(k)κ⎞ ⎠≤Kmax(1, (s+rs+1+···+rd)κ), (5.9) where K=n−1 j=d+1|yj|and κ=maxn−1 j=d+1d k=1νjk.Finally,usingthat(−uj)≤ρ<for j=1, ...,, (−uj)=−rjfor j=s+1, ...,dand (5.8), we obtain M(u,y)≤s−rs+1−···−rd+|yn||a|+Kmax(1, (s+rs+1+···+rd)κ). (5.10) Since rs+1+···+rd≤u1=ρs+rs+1+···+rd≤s+rs+1+···+rdfor u∈Υ() η, we still get an inequality of type (3.10). There are constants Cη,cη>0 (depending also on ybut independent of ) such that M(y,u)≤Cη−cηu1for all u∈Υ() η. A closer look at the argument that proves (5.10) shows that we can write the following upper bound for |M(u,y)|: |M(u,y)|≤d+|yn||a(n)|+K(d+u1)κ. This upper bound, the relation (5.10) in the form of inequality M(y,u)≤Cη−cηu1 and the fact that 0 <κ<1 prove that M(u,y)/M(u,y)tends to zero as u1tends to infinity. In particular the argument of M(u,y)tends to πalong Υ() η. We need a similar result when η=∅.WenoticethatM(u,y)is univalent on the 2dbranches of Υ() ∅. Following the proof of inequality (5.10) we obtain the inequality: M(u,y)≤−r1−···−rd−|yn|ra(n)+Kmax(1, (r1+···+rd)κ). Since for the imaginary part we have the inequality |M(u,y)|≤|yn|ra(n)+ K(u1)κ, we deduce that if we set αn=|π−arg yn|∈[0, π/2[, we have for any δ>0and any u1large enough arg M(t,y)∈]π−αn−δ,π+αn+δ]. Let us use a good compactification Xof (C∗)d, a real blow-up π: X−→Xof X along D, and apply Corollary 5.2. The behaviour of arg M(t,y)when u1→+∞and the fact that for any R>,Υ()∩{u|u1≤R}is compact imply that Υ() is a rapid decay cycle. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17857 Let us prove that when β<0, the integral HΥ()(β;y)tends, when →0, to the integral (5.4) multiplied by the factor T:= ξ∈{0,1}d (−1)d−|ξ|exp 2√−1πβkqkξk= d k=1 (exp(2√−1πqkβk)−1)(5.11) that comes from the parametrization (5.7)forη=∅.Since(5.4) is clearly the limit of the piece of the integral HΥ()(β;y)over Υ() ∅, it suffices to show that the integrals over Sη for η= ∅tend to zero. Let us assume again for simplicity that η={1, ...,s}with 1 ≤s≤d. On each piece of Υ() ηthe parameters are (θ1,...,θs,rs+1,...,rd)∈ k∈η [0, 2qkπ]×Uη. and the change of variables from the parametrization (5.7) induces the following results in the different factors of the integrand: d k=1 duk uk=(√−1dθ1)∧···∧(√−1dθs)∧drs+1 rs+1∧···∧drd rd , u−β=ρ−β1−···−βsr−βs+1 s+1···r−βd dexp ⎛ ⎝√−1⎛ ⎝− s j=1 βjθj− d k=s+1 2πβkqkξk⎞ ⎠⎞ ⎠, |u−β|=ρ−(β1+···+βs) d =s+1 r−β exp ⎛ ⎝ s j=1βjθj+ d k=s+1 2πβkqkξk⎞ ⎠ ≤−(β1+···+βs) d =s+1 r−β exp d k=1 2π|βk|qk. From these inequalities and the fact that the real part of the argument of the exponential function is bounded from above by Cη−cη(rs+1+···+rd) with Cη,cη∈R>0independent of ,for∈[0, 0], we see that the integral over Υ() η tends to zero when →0 as expected, because −(β1+···+βs)>0. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
17858 F.-J. Castro-Jiménez et al. Finally, let us prove that the integral HΥ() does not depend on : take 0 < 1< 2. We consider Υ([1,2]), the noncompact (d+1)-cycle ∈[1,2]{}×Υ() with oriented boundary {1}×Υ( 1)−{2}×Υ( 2). Consider then for R> 2the compact cycle ΥR=Υ([1,2])∩([1,2]×PR), where PRis the polydisk PR={u∈Cd||u1|≤R,...,|ud|≤R}. Integrals HΥ() are of the form HΥ() =Υ()ω, where ωis a holomorphic d–form independent of and hence it is a closed form. We have 0=ΥR dω=∂ΥR ω. The boundary ∂ΥRis equal to ({1}×Υ( 1)) ∩([1,2]×PR)−({2}×Υ( 2)) ∩([1,2]×PR)+∂R. Since by examining the parametrization (5.7) we see that each d-dimensional piece of ∂R is included in an hyperplane uj=R, hence, the restriction to it of ωis zero. We deduce that the integral of ωon Υ( j)∩PR(which can replace ({j}×Υ( j))∩([1,2]×PR)because ωdoes not depend on )forj=1, 2 are equal. Taking the limit when R−→∞ we obtain the result HΥ( 1)=HΥ( 2). In the case of general p,δ, we keep the same cycle and work with the integral HΥ()(β;y):=Υ() u−β−1exp ⎛ ⎝− d k=1 e√−1πδkuk+ n j=d+1 zjua(j)⎞ ⎠du, where zj=e√−1π1+2p+δ,a(j)yjand the proof is essentially the same with only an easy modification of inequality (5.10). In particular, the cycle Dp,δin the statement of Theorem 5.3, is the image of Υ() by tk=uk·exp(√−1π1+2p+δ,a(k)). Conclusion: The integral HΥ()(β;y)is analytic as a function of β∈Cd. Moreover, when β<0, e−√−1π1+2p+δ,βHΥ()(β;y)=T·Fp,δ(β;y), see (5.11). Hence, it equals the Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
Gevrey Expansions of Hypergeometric Integrals II 17859 meromorphic continuation T· Fp,δ(β;y)outside the union of hyperplanes Pdescribed in Lemma 4.9. When qkβk/∈Zfor all k∈{1, ...,d}, the factor Tis non zero and we obtain a Gevrey series expansion for the integral along rapid decay cycles HΥ()(β;y). To check this last claim we have to remark that the set of poles of the analytic continuation Fp,δ(β;y)is contained in Pwhich is contained in the set defined by T=d k=1(exp (2√−1πqkβk)−1)=0. This latter set is, under Assumption 4.1, the set of parameters β=B−1 σγsuch that γis called resonant for B(see [12, 2.9]). Coming back to the general situation of Theorem 4.3, the result of this theorem and the above considerations prove the following theorem: Theorem 5.8. If Assumption 4.1 is satisfied and γ∈Cdis non resonant for B,then all the Gevrey solutions of MB(γ ) along the hyperplane xn=0 can be described as linear combinations of a fixed set of asymptotic expansions of integral solutions of type IC(γ ;x)along rapid decay cycles. Funding This work was supported by Ministerio de Ciencia, Innovación y Universidades MTM2016-75024- P, FEDER, and FQM333-Junta de Andalucía [to F.-J.C.-J. and M.-C.F.-F.]. Acknowledgments We would like to thank K. Takeuchi and S.-J. Matsubara-Heo for their suggestions and useful comments about the content of this article. We also thank two anonymous referees for their careful reading and helpful suggestions. The 1st author would like to thank the Département de Mathématiques of the University of Angers (France) for its support during the 1st stage of this research. The 3rd author would like to thank the Department of Algebra and the Institute of Mathematics of the University of Seville (IMUS) for their support and hospitality during the preparation of this paper. References [1] Adolphson, A. “Hypergeometric functions and rings generated by monomials.” Duke Math. J. 73, no. 2 (1994): 269–90. [2] Aomoto, K. and M. Kita. “Theory of hypergeometric functions.” Springer Monogr. Math., Springer-Verlag, Tokyo, 2011. [3] Assi, A., F.-J. Castro-Jiménez, and M. Granger. “The Gröbner fan of an an-module.” J. Pure Appl. Algebra 150, no. 1 (2000): 27–39. Downloaded from https://academic.oup.com/imrn/article/2021/23/17823/5648064 by Universidad de Sevilla user on 28 June 2022
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