Irregular modified A-Hypergeometric systems

arXiv:1207.1533v2 [math.CA] 3 Jul 2013 Irregular Modified A-Hypergeometric Systems Francisco-Jes´ us Castro-Jim´ enez∗ , Mar´ ıa-Cruz Fern´ andez-Fern´ andez∗, Tatsuya Koike and Nobuki Takayama Monday 22nd July, 2013 Abstract A modified A-hypergeometric system is a system of differential equations for the function f(tw·x)where f(y)is a solution of an A-hypergeometric system in nvariables and w is an ndimensional integer vector, which is called the weight vector. We study the irregularity of modified systems by adapting to this case the notion of umbrella introduced by M. Schulze and U. Walther. Especially, we study slopes and Gevrey series solutions. We develop some applications of this study. Under some conditions we give Laplace integral representations of divergent series solutions of the modified system and we show that certain Gevrey series solutions of the original A-hypergeometric system along coordinate varieties are Gevrey asymptotic expansions of holomorphic solutions of the A-hypergeometric system. Contents 1 Introduction 2 2 Generalities on slopes 5 3 On the irregularity of A–hypergeometric systems 6 3.1 Slopes of A–hypergeometric systems . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Gevrey solutions of A–hypergeometric systems at infinity. . . . . . . . . . . . . . 7 4 On the irregularity of modified A-hypergeometric systems 11 4.1 Fourier transform and initial ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Slopes of modified A-hypergeometric systems . . . . . . . . . . . . . . . . . . . . 12 4.3 Holomorphic solutions of a modified hypergeometric system . . . . . . . . . . . . 15 4.4 Gevrey solutions of a modified hypergeometric system . . . . . . . . . . . . . . . 16 4.5 Gevrey solutions modulo convergent series . . . . . . . . . . . . . . . . . . . . . 19 ∗First two authors are partially supported by MTM2010-19336 and FEDER and Junta de Andaluc´ıa FQM5849, FQM333. FJCJ is also partially supported by S-13025-JSPS (Japan); MCFF is also partially supported by Max Planck Institute f¨ur Mathematik (Bonn). Third author is partially supported by JSPS grants-in-aid No. 21740098 and No. S-24224001. 1 5 Borel transformation and asymptotic expansion 21 6 Borel transformation revisited 30 1 Introduction A-Hypergeometric systems (or GKZ-systems or simply hypergeometric systems) are systems of linear partial differential equations on the complex affine space Cn. Although they were already considered in works by J. Hrabowski [19] and by I. M. Gelfand, M. I. Graev and A. V. Zelevinsky [13], the systematic study of hypergeometric systems started with the paper by I. M. Gel’fand, M. M. Kapranov and A. V. Zelevinsky [14]. Each of these systems, denoted by HA(β), is determined by a pair (A, β)where A= (aij)is an integer d×nmatrix of rank dand β∈Cd is a parameter vector. An A-hypergeometric system HA(β)is defined by the dEuler operators Ei−βi:= Pn j=1 aijyj∂ ∂yj−βifor i= 1,...,dand the toric operators ∂u−∂vassociated with each pair (u, v)∈Nn×Nnsuch that Au =Av. Here ∂ustands for the monomial differential operator ∂u1 1···∂un nand ∂i=∂ ∂yi. For generic parameters β∈Cd, the holomorphic solutions of HA(β)at nonsingular points can be described by using the so-called Γ–hypergeometric series (see [13], [14]; see also [26] and Subsection 3.2). We are interested in divergent Γ–hypergeometric series solutions. To study them we use the notion of slopes defined in the general setting by Y. Laurent [21]. If the matrix Ais pointed (which means that the column vectors of Alie in an open half-space with boundary passing through the origin in Rd), M. Schulze and U. Walther [27] have described the slopes of HA(β)with respect to coordinates varieties, generalizing previous work in [7], [16] and [17]. These slopes are closely related to the irregularity of the system [22] and the existence of non convergent Gevrey series solutions of this system. Gevrey series solutions of HA(β)were studied by the second author in [10] (see also [11] and [12] where the authors treat particular cases). Modified hypergeometric systems were introduced by the fourth author [30] in order to study solutions of hypergeometric systems along a curve y(t) = (c1tw1,...,cntwn)for w∈Zn,ci∈ C. Each of them is determined by a tuple (A, w, β, α)where Aand βare as before, w= (w1,...,wn)∈Znand α∈C. We denote by e A(w)(or simply e A) the matrix e A(w) = e A=    a11 ··· a1n0 ··· 0 adn ··· adn 0 w1··· wn1    . Throughout this paper, we do not always assume that Ais pointed, but we assume that e Ais. Note that when Ais pointed, then e Aalso is. Definition 1 ([30]) We call the following system of differential equations HA,w,α(β)amodified 2 A-hypergeometric system: n X j=1 aijxj∂j−βi!•f= 0,(i= 1,...,d)(1) n X j=1 wjxj∂j−t∂t−α!•f= 0,(2) n Y i=1 ∂ui itun+1 − n Y j=1 ∂vj jtvn+1 !•f= 0 (3) with u, v ∈Nn+1 running over all u, v such that e Au =e Av. Here we denote ∂ ∂xi=∂i. The modified system is defined on the space X=Cn+1 with coordinates (x, t) = (x1,...,xn, t). Let D(or Dn+1) be the Weyl algebra in (x, t). The left ideal in Dgenerated by the operators in (1), (2), and (3) is also denoted by HA,w,α(β)if no confusion arises. The left D-module D/HA,w,α(β) is denoted by MA,w,α(β). By [30] the D-module MA,w,α(β)is holonomic for any A, β, w, α. The system HA(β)is a summand of the modified system on the space t6= 0. More precisely, denote Y=Cn+1 with coordinates (y, s)and consider the map ϕ:Y∗:= Cn×C∗⊂Y−→ X∗:= Cn×C∗⊂X(4) defined by ϕ(y1,...,yn, s) = (s−w1y1,...,s−wnyn, s). The pullback image of the ideal HA,w,α(β) by ϕequals the ideal of differential operators on Y∗generated by HA(β)and s∂s+α. Notice that this last ideal is nothing but the hypergeometric ideal associated with the matrix e A(0)and the parameter vector (β, −α)∈Cd+1. As usual we denote this ideal by He A(0)(β, −α). Let us consider the local analytic situation. Let Dbe the sheaf of holomorphic differential operators on Xand MA,w,α(β)the quotient sheaf D/DHA,w,α(β). By the previous observation, the hypergeometric D-module Me A(0)(β, −α) = D/DHe A(0)(β, −α)is the extension to Yof the pullback module ϕ∗(MA,w,α(β)) considered as a DY∗–module on the first space Y∗=Cn× C∗. Since ϕis a biholomorphic map both D-modules Me A(0)(β, −α)|Y∗and MA,w,α(β)|X∗are isomorphic. We can describe solutions of the original hypergeometric system HA(β)associated to the weight vector w∈Znvia solutions of the modified system. Series solutions of HA(β)have been studied in [13] and [14] where the authors constructed convergent series solutions associated to the regular triangulation induced by a generic weight vector w. The construction is generalized as follows [26]: Assume that β∈Cdis very generic (this condition is essential in the construction). Suppose that the initial ideal in(−w,w)(HA(β)) has a solution of the form yρ,ρ∈Cn. Then the monomial yρcan be extended to a formal series solution φ(y) = yρ+··· of HA(β). We call the series φ(y)a series solution of HA(β)associated to the weight vector w. The series is divergent in general. We are interested in giving an explicit expression of a solution of HA(β)whose asymptotic expansion is φ(y). A standard method, in the theory of ordinary differential equations, to construct such an expression is the Laplace integral representation and the Borel transformation of divergent series. This method has been successful in the study of global analytic properties of solutions of ordinary differential equations, see, e.g., the book by W. Balser [3] and the references 3 therein. Then it is a natural problem to construct a Laplace integral representation corresponding to the divergent series solution φ(y)associated to the weight vector w. Our modified system, which is a system of differential equations for φ(tw1x1,...,twnxn), is used to give an answer to this problem. A key ingredient of our study is the fact that the modified hypergeometric system HA,w,α(β)is transformed into the hypergeometric system He A(w)(β, α −1) associated with the matrix e A(w)and parameter (β, α −1), by the formal inverse Fourier transform t7→ ∂t,∂t7→ −t, which enables us to study the modified system using the theory of hypergeometric systems (see Subsection 4.1). For example, for A= (1,2) and β∈Cthe system HA(β)is generated by x1∂1+ 2x2∂2− βand ∂2 1−∂2. The modified system HA,w,α(β)for w= (−1,−1),α∈Cis generated by the three operators x1∂1+ 2x2∂2−β, −x1∂1−x2∂2−t∂t−α, ∂2 1t−∂2, and the inverse Fourier transformation of HA,w,α(β)is generated by x1∂1+2x2∂2−β, −x1∂1−x2∂2+t∂t−α+1, ∂2 1∂t−∂2 which is equal to He A(w)(β, α −1). We study the behavior of solutions of HA,w,α(β)near the hyperplane t= 0 in the space Xand we will give Laplace integral representations of its solutions. The structure of the paper is as follows: in Section 2 we recall Y. Laurent’s definition of the slopes of a finitely generated D-module with respect to a hypersurface. In Section 3 we recall the use of umbrellas for the description of the slopes of a hypergeometric system given by Schulze and Walther [27], then we summarize the construction of the Gevrey solutions given in [10] and extend some of these results to the case of the Gevrey solutions at infinity. In Section 4 we provethat the formal inverseFourier transform, with respect to T, of a modified hypergeometric system is an A-hypergeometric system and we use this fact to describe the slopes of the former by using the umbrella of the latter. We construct, associated with any slope of the modified system MA,w,α(β), a basis of its Gevrey solutions, modulo convergent power series, when the parameters βand αare very generic, see Theorem 6. Moreover, for β∈Cdvery generic we construct a basis of formal series solutions of the modified system for any α∈Cand w∈Zn, see Theorem 5. If in addition wis generic this basis is reduced to a single element. Later, in Sections 5 and 6, we prove under some assumptions that this solution is an asymptotic expansion of a Laplace integral representation of a solution by the Borel summation method (Theorem 8 and Section 6). As an application, we give an asymptotic error evaluation of finite sums of formal series solutions of original A-hypergeometric systems with irregular singularities studied in, e.g., [10] and [14] (see Theorem 8, the inequality (17), and Section 6). Example 4 illustrates the application for the simplest A. Several integral representations have been studied for solutions of regular holonomic hypergeometric systems (see, e.g., [15], [6], [4]). They play a prominent role in the study of Ahypergeometric functions. However, there have been few studies of integral representations for solutions of irregular A-hypergeometric systems. We would like to point out that integral representations of holomorphic solutions of irregular A-hypergeometric systems have been recently given by A. Esterov and K. Takeuchi [9] by using the so called rapid decay homology cycles. Our Laplace integral representation, which we propose in this paper for giving an analytic meaning to divergent series solutions, is different from their representation: the integrandof our representation is an A-hypergeometric function associated to a homogenized configuration of A(Section 5). InSection 6 weillustratehow, under someconditions, thestudyof theirregularityofMA,w,α(β) 4 along Tgives an analytic meaning to the Gevrey series solutions of MA(β), along coordinate varieties, constructed in [10]. More precisely we prove (see Proposition 5) that they are asymptotic expansions of certain holomorphic solutions of MA(β). An interplay of algebra (slopes and formal series) and analysis (Borel summation method) is a main point of this paper. In order to make a comprehensible presentation to readers from several disciplines, we often review some well-knownfacts to experts. We hope that our style is successful. Acknowledgements: We wish to thank J. Gonz´alez-Meneses, M. Granger and D. Mond for their help, suggestions and comments. We are very grateful to an anonymous referee, whose thoughtful suggestions have improved this article. 2 Generalities on slopes Recall that D=Dn+1 is the Weyl algebra Chx1,...,xn, t, ∂1,...,∂n, ∂ti. In this section we write x= (x1,...,xn+1),∂= (∂1, . . . , ∂n+1). The variable tis also denoted by xn+1 and ∂tby ∂n+1. Let L:R2n+2 →Rbe a linear form L(α, β) = Piuiαi+viβisuch that ui+vi≥0for i= 1,...,n+ 1, inducing the so-called L–filtration on the ring D. If ui+vi>0for all i, the associated graded ring grL(D)is isomorphic to a polynomial ring in 2n+ 2 variables (x, ξ) = (x, ξ1,...,ξn+1)with complex coefficients. This polynomial ring is L-graded, the L-degree of a monomial xαξβbeing L(α, β). If we need to emphasize the coefficients of the linear form we simply write L=L(u,v)for (u, v)∈R2n+2 with ui+vi≥0for all i. If u=0∈Nn+1 and v=1= (1,1,...,1) ∈Nn+1 then the corresponding L(u,v)filtration is nothing but the usual order filtration on D(which is also called the F-filtration). If u= (0,−1) ∈Nn+1 and v=−u∈Nn+1 then the corresponding L(u,v)filtration is nothing but the Malgrange-Kashiwara filtration on D (also known as the V-filtration) with respect to t= 0. In the remainder of this section we assume ui+vi>0for all i; we say then that (u, v)is a weight vector for the Weyl algebra D. All the D–modules appearing here are left D-modules unless stated otherwise. We denote by T⊂Cn+1 the hyperplane defined by t= 0. Let Mbe a finitely generated D-module. To the L-filtration on Dwe associate a good Lfiltration on M, by means of a finite presentation. The associated grL(D)-module grL(M)is then finitely generated. The radical of the annihilating ideal AnngrL(D)(grL(M)), which is independent of the good L–filtration on M, defines an affine algebraic subset of the cotangent space T∗Cn+1 = C2n+2. This algebraic set is called the L-characteristic variety of Mand it is denoted by ChL(M). The results stated so far are well known in D-module theory and generalize [5] which treats the case of the F–filtration in D. The case of a general L–filtration has been studied for example in [20] and, with more details, in [21, Section 3.2] in the micro-differential setting which is slightly different from the one in this paper. See also [2, Section 2] for an equivalent treatment better adapted to effective computations for modules on the Weyl algebra. We consider a special type of L-filtration: For any real number r∈R≥0we denote by Lreither the linear form Lr=F+rV or the filtration on Dgiven by the (2n+ 2)–dimensional weight vector (0,...,0,−r, 1,...,1,1+r)where −ris placed in the (n+1)th-component. Here F(resp. V) stands for the order filtration on D(resp. the Malgrange-Kashiwara filtration with respect to T). 5 Definition 2 [21, Section 3.4] Let Mbe a finitely generated D–module. Consider the projection Π : T∗Cn+1 −→ Tdefined by Π(x1,...,xn, t, ξ1,...,ξn, ξt) = (x1,...,xn,0). For any real number r > 0, let Ir T(M)be the closure of the projection by Πof the irreducible components of the Lr–characteristic variety ChLrM⊂T∗Cn+1 that are not (F, V )–bihomogeneous. The real number s=r+ 1 >1is said to be a slope of Malong Tat p∈Tif and only if p∈Ir T(M). As proved by Y. Laurent, see [21, Section 3.4], any slope is a rational number and the set of slopes of Mis finite. Moreover, Y. Laurent also proved loc. cit. that s=r+ 1 is a slope of M along Tat p∈Tif and only if, in a neighborhood of Π−1(p),ChLr′(M)is not locally constant for r′∈(r−ǫ, r +ǫ)with ǫ > 0small enough. The irregularity of a holonomic system with respect to a smooth hypersurface ([24, D´ef. 6.3.1]) is deeply related with the slopes of the system defined with respect to the given hypersurface [22]. 3 On the irregularity of A–hypergeometric systems Recall that Ais a d×ninteger matrix of rank dwhose columns a1,...,angenerate Zdas Z– module. We denote by HA(β)the hypergeometric ideal associated with Aand the parameter vector β∈Cd[14] and by MA(β)the corresponding hypergeometric system (also known as GKZ–system). This system is the quotient of Dn:= C[x1,...,xn]h∂1,...,∂ni, the Weyl algebra of order n, by the left ideal HA(β). In this section we denote X=Cn. 3.1 Slopes of A–hypergeometric systems We recall in this subsection some results from [27], where Ais assumed to be pointed, i.e. the columns of Alie in a open half-space defined by a hyperplane passing through the origin in Rd. These results will not be applied to our matrix Abut only to the matrix e A(w)(see Subsection 4.2), which we assume to be pointed throughout this article. We denote by aithe i-th column of Afor i= 1,...,n. The L-characteristic variety of the hypergeometric system MA(β)has been described, in a combinatorial way, by M. Schulze and U. Walther [27] for any pointed matrix Aand any filtration L= (u, v)such that ui+vi=c > 0 for all i= 1,...,n. The case L=Fwas first studied by A. Adolphson [1]. The main tool for their description is the notion of (A, L)-umbrella that we define here for the sake of completeness. First of all, the (A, L)–umbrella only depends on Aand on the coefficients viof the linear form L=L(u,v). Definition 3 [27, Def. 2.7] We assume that vi>0for all i. The (A, L)–polyhedron ∆L Ais the convex hull in Rdof the set {0, a1/v1,...,an/vn}. The (A, L)-umbrella ΦL Ais the set of faces of ∆L Awhich do not contain zero. In particular, ΦL Acontains the empty face. By ΦL,k A⊂ΦL Awe denote the subset of faces of dimension k. We identify each face σof ΦL A with the set {i:ai/vi∈σ}and with {ai:ai/vi∈σ}. The (A, L)-umbrella is then an abstract cell complex. 6 When not all the viare strictly positive then both definitions of the (A, L)-polyhedron and the (A, L)-umbrella are a little bit more involved. We refer to [27, Def. 2.7] for these precise definitions in the general case. See also Subsection 4.2. Theorem 1 [27, Cor. 4.17] The L–characteristic variety of MA(β)is given by: ChL(MA(β)) = [ τ∈ΦL A Cτ A(5) where Cτ Ais the closure of the conormal Cτ Aof the torus orbit Oτ A={ξ∈T∗ 0X=Cn:ξi= 0if i /∈τ, ξi=yaiif i∈τ, y ∈(C∗)d}. Moreover, it is proved in [27, Lemma 3.14] that Cτ Ameets T∗ 0Xfor all τ∈ΦL A. Thus Theorem 1 provides a description of the slopes of a hypergeometricsystem at the origin along any coordinate variety Y⊂Xvia considering the 1–parameter family of filtrations Lr=F+rV ,r > 0, where Vis the V-filtration1along Y: Corollary 1 [27, Cor. 4.18] The real number s=r+ 1 >1is a slope of MA(β)along Yat the origin if and only if ΦLr′ Ais not locally constant at r′=r. Remark 1 When Y⊂Xis a coordinate hyperplane then the set of slopes of MA(β)along Yat the origin coincides with the set of slopes of MA(β)along Yat any point p∈Y. This is proved in [10, Th. 5.9] by using the comparison result in [22, Th. 2.4.2]. 3.2 Gevrey solutions of A–hypergeometric systems at infinity. In this subsection we extend some of the results from [10] to the case of the Gevrey solutions of a hypergeometric system at infinity in the direction of a coordinate hyperplane that we may assume to be xn= 0. This construction is used later in the study of the Gevrey solutions along Tof a modified hypergeometric system, see Subsection 4.5. Let us denote by OXthe sheaf of holomorphic functions on X=Cn. For Y={xn= 0}, we denote by Od X|Ythe formal completion of OXalong Y, whose germs at (p, 0) ∈Yare of the form f=Pm≥0fmxm nwhere all the fm=fm(x1,...,xn−1)are holomorphic functions in a common neighborhood of p. Notice that the restriction of OXto Y, denoted by OX|Y, is a subsheaf of Od X|Y. For any real number s, we also consider the sheaf Od X|Y(s)of Gevrey series along Yof order s which is defined to be the subsheaf of Od X|Ywhose germs fat (p, 0) ∈Ysatisfy X m≥0 fm m!s−1xm n∈ OX|Y,(p,0). We denote QY(s) := Od X|Y(s) OX|Yand use Od X|Y(<s)for the sheaf of Gevrey series along Yof order less than s. If fbelongs to Od X|Y(s)\ O d X|Y(<s)for some s, we say that the index of the Gevrey series fis s. We also write Od X|Y(+∞) := Od X|Y. 1The V–filtration with respect to the coordinate variety Y= (x1=··· =xℓ= 0) is defined by assigning the weight -1 (resp. the weight 1) to the variablesxi(resp. ∂i) for i= 1,...,ℓand the weight 0to the remainingvariables. 7 We denote by D=DXthe sheaf of linear differential operators on Xwith holomorphic coefficients. If Mis a coherent D-module, Z. Mebkhout has defined in [24, D´ef. 6.3.1] the irregularity of order sof Mwith respect to Yto be the complex of (sheaves of) vector spaces Irr(s) Y(M) := RHomD(M,QY(s)) and has proved that, for all s∈[1,+∞], this complex is a perverse sheaf on Ywhen Mis holonomic [24, Th. 6.3.3]. By the comparison theorem [22, Th.2.4.2] sis a slope of Mwith respect to Yif and only if sis a gap in the filtration Irr(s) Y(M) on the irregularity IrrY(M) := Irr(+∞) Y(M). The perversity result implies, in particular, that at a generic point p∈Yonly the first cohomology space of previous complexes is possibly non zero and thus it is worth studying the stalk HomD(M,QY(s))p. Recall that A= (a1··· an)is a full rank matrix with ai∈Zdfor all i= 1,...,nand d≤n. Following [14] and [26], for any vector v∈Cnwe can define a series φv=φv(x) := X u∈Nv [v]u− [v+u]u+ xv+u(6) where v∈Cnverifies Av =βand Nv={u∈ker(A)∩Zn: nsupp(v+u) = nsupp(v)}. Here ker(A) = {u∈Qn:Au = 0},nsupp(w) := {i∈ {1,...,n}:wi∈Z<0}is the negative support of w∈Cn,[v]u=Qi[vi]uiand [vi]ui=Qui j=1(vi−j+ 1) is the Pochhammer symbol for vi∈C,ui∈N. The series φvis annihilated by the hypergeometric ideal HA(β)if and only if the negative support of vis minimal, i.e., ∄u∈ker(A)∩Znwith nsupp(v+u)(nsupp(v)(see [26, Section 3.4]). When β∈Cdis very generic, i.e., when βis not in a locally finite countable union of Zarisky closed sets, there is a basis of the Gevrey solution space of MA(β)along Ygiven by series φvfor suitable vectors v∈Cn(see [10, Ths. 6.2 and 6.7]). For any subset η⊆ {1, . . . , n}we denote by Aηthe submatrix of Agiven by the columns of A indexed by ηand we denote ¯η={1,...,n} \ η. We say that σ⊆ {1,...,n}is a (d−1)-simplex with respect to A(or simply that σis a (d−1)- simplex) if the columns of Aσdetermine a basis of Rd. If it is so, we can reorder the variables in order to have σ={1,...,d}without loss of generality. Then a basis of ker(A)associated with σ is given by the columns of the matrix: Bσ=       −A−1 σad+1 −A−1 σad+2 · · · −A−1 σan 1 0 0 0 1 0 . . ..... . . 0 0 1        A vector v∈Cnsatisfying vi∈Nfor all i /∈σand Av =βcan be written as vk= (A−1 σ(β−X i/∈σ kiai),k) for some k= (ki)i/∈σ∈Nn−d. Since βis very generic then the negative support of vkis the empty set and hence φvkis annihilated by HA(β). Moreover, the summation index set Nvkin the series 8 φvkis given by the integer vectors in an affine translate of the positive span of the columns of Bσ. The sum of the coordinates of the i-th column of Bσis 1− |A−1 σad+i|where | | means the sum of the coordinates. We have the following. Theorem 2 [10, Theorem 3.11] Under the above conditions the series φvkis a Gevrey solution of MA(β)along Z={xj= 0 : |A−1 σaj|>1}with index s= max{|A−1 σaj|:j= 1,...,n}at points in certain relatively open subset of Z. In particular, if |A−1 σaj| ≤ 1for all 1≤j≤nthen φvkis convergent. By Corollary 1, a real number s > 1is a slope of MA(β)along Y={xn= 0}if and only if 1 san belongs to the hyperplane Hτsupported on a facet τof the convex hull of {0, a1,...,an−1}such that 0/∈τ. In particular, for any (d−1)-simplex σ⊆τit is easy to check that s=|A−1 σan|>1. We say in this case that σis a (d−1)-simplex corresponding to the slope s > 1of MA(β)along Y.The following Theorem is a summary of some of the results from [10]. Its last statement uses results from [22] and in [27]. For the definition of a regular triangulation see, e.g., [29, Ch. 8]. Theorem 3 Assume that β∈Cdis very generic and that s > 1is a slope of MA(β)along Y={xn= 0}. For any (d−1)-simplex σcorresponding to sone can construct vol(σ) = |det(Aσ)|many linearly independent Gevrey solutions φvkof MA(β)along Ywith index sby varying k∈Nn−din a set Λso that {Aσk|k∈Λ}is a set of representatives of the group Zd/ZAσ. Moreover, if we repeat this construction for all the (d−1)-simplices σcorresponding to swhich belong to a suitable regular triangulation for Aand take the classes modulo Od X|Y(<s)we obtain a basis for the space of solutions of MA(β)in the space (Od X|Y(s)/Od X|Y(<s))pfor points p∈Y in a relatively open set of Y. We have exhibited the construction of the Gevrey solutions of MA(β)along Y={xn= 0} corresponding to each slope s > 1of MA(β)along Yfor βvery generic. Let us construct Gevrey solutions of MA(β)at infinity. In other words, we are going to construct Gevrey solutions of the projectivized hypergeometric system treated in [28, Section 5] at a generic point at infinity. We use the followingnotation: X′is Cnwith coordinates(x1,...,xn−1, z) and z= 1/xnso that X∩X′=Cn−1×C∗. Denote Y′={xn=∞} ={z= 0} ⊆ X′. Take L−r=F−rV where Vis the V-filtration along Y. Notice that ΦL−r Ais not locally constant at r=s−1>0if and only if 1 (2−s)anbelongs to the hyperplane Hτsupported on a facet τof the convex hull of {0, a1, . . . , an−1}such that 0/∈τ. Theorem 4 Assume that β∈Cdis very generic and that there exists s > 1such that 1 (2−s)an belongs to a hyperplane Hτas above. For any (d−1)-simplex σ⊆τone can construct vol(σ) = |det(Aσ)|many linearly independent Gevrey series φvkalong Y′with index sby varying k∈ Nn−d−1×Z<0in a set Λso that {Aσk:k∈Λ}is a set of representatives of the group Zd/ZAσ. The classes of these series modulo convergent series OX′|Y′are solutions of MA(β) in O\ X′|Y′(s)/OX′|Y′. Moreover, if we repeat this construction for all the (d−1)-simplices σas above which belong to a suitable regular triangulation for Aand take the classes modulo O\ X′|Y′(<s)then we obtain a basis for the space of solutions of MA(β)in the space (O\ X′|Y′(s)/O\ X′|Y′(<s))pfor points p∈Y′ in a relatively open set of Y′. 9 4.4 Gevrey solutions of a modified hypergeometric system We describe the solutions of MA,w,α(β)in the space Od X|Tof formal power series with respect to T={t= 0} ⊂ X=Cn+1. More generally, we also describe the solutions of MA,w,α(β)in the space Pγ∈ΛtγOd X|Tfor any finite set Λ⊆C(see Theorem 5). We will use notations in [30]. Let τbe the weight vector (0,−1,0,1) ∈Z2n+2 inducing the Malgrange-Kashiwara V–filtration along T≡(t= 0) on the ring Dn+1. Let ˜ Ie A(w)⊆C[∂, t] := C[∂1,...,∂n, t]be the toric ideal associated with e A(w), i.e., the binomial ideal generated by the operators in (3). Lemma 3 For all w∈Znwe have in(0,−1)(e Ie A(w)) = C[∂, t]inw(IA). Proof. Recall that e Ie A(w)=h∂u+−∂u−|Au = 0, w ·u= 0i+h∂u+−tw·u∂u−|Au = 0, w ·u > 0i and we can write IA=h∂u+−∂u−|Au = 0, w ·u= 0i+h∂u+−∂u−|Au = 0, w ·u > 0i. Notice that in(0,−1)(∂u+−tw·u∂u−) = ∂u+= inw(∂u+−∂u−)if Au = 0 and w·u=w·u+−w·u−> 0and that in(0,−1)(∂u+−∂u−) = ∂u+−∂u−= inw(∂u+−∂u−)if Au = 0 and w·u= 0. The conclusion follows by a straightforward Groebner basis argument because a Groebner basis of e Ie A(w)with respect to (0,−1) (resp. of IAwith respect to w) is given by a set of binomials with the same form as the ones defining the ideal. Q.E.D. Recall that the indicial polynomial (also called b-function) of HA,w(β)along Tis the polynomial b(s)∈C[s]such that b(θt)is the monic generator of inτ(HA,w(β)) ∩C[θt]where θt=t∂t. Moreover, we have by [30, Th. 3] that for βand wgeneric, the indicial polynomial of HA,w(β) along Tis b(s) = Y (∂k,σ)∈T (M) (s−wβ(∂k,σ))(10) where M= inw(IA),T(M)is the set of top-dimensional standard pairs of M(see [26, Sec. 3.2]) and v=β(∂k,σ)is the vector defined as vi=ki∈Nfor i /∈σand Av =β, which is also an exponent of HA(β)with respect to w(see [26, Lemma 4.1.3]). Definition 5 We say that a (generic) vector ew∈Qnis a (generic) perturbation of w∈Zn, with respect to A, if there exists w′∈Qnsuch that inew(IA) = inw′(inw(IA)). Remark 7 If ewis generic then inew(IA)is a monomial ideal and it is well known that its degree equals the cardinality of its set of top-dimensional standard pairs T(inew(IA)). Moreover, for very generic β∈Cdthere are exactly deg(inew(IA)) many exponents of HA(β)with respect to ew; see [26, Sec. 3.4] and [8, Prop. 4.10]. Lemma 4 Let β∈Cdbe very generic and w∈Zn. There is a generic perturbation ew∈Qnof w such that for any exponent v∈Cnof HA(β)with respect to ewthe series ψv(x, t) = t−αφv(twx), for twx= (tw1x1,...,twnxn), is a solution of MA,w,α(β)of the form ψv(x, t) = Pm≥0fm(x)tγ+m∈ tγOd X|T,(p,0), with γ=wv −αand f0(x)6= 0 for some p∈Cn. 16 Proof. Since βis very generic, for any generic ewan exponent vof HA(β)with respect to ew can be written as v=β(∂k,σ)where (∂k, σ)is a top-dimensional standard pair of inew(IA), see [26, Sec. 3.4]. In particular, σ∈Φew,d−1 A,k= (ki)i6∈σ∈Nn−d,Av =βand vi=ki∈Nfor all i /∈σ. The series φv(x)is either a holomorphic solution or a Gevrey solution of MA(β)along a coordinate subspace Z⊆Cnat any point pin a non empty relatively open set Uσin Z(see Theorem 2 and [10, Th. 3.11] for the details) and since βis very generic we have that Nv= (−Bσk+NBσ)∩Zn. The expression f(x, t) := t−wvφv(tw1x1,...,twnxn) = tα−wvψv(x, t)(resp. ψv(x, t)) formally satisfies the equations defining MA,w,wv(β)(resp. MA,w,α(β)). We will prove that we can write f(x, t) = Pm≥0fm(x)tmand that it is a Gevrey series along T⊂X. Recalling the expression of φv(6) it is enough to prove that for all u∈Nv\ {0}we have wu ∈Nand that the coefficient of tmin f, i. e. fm(x) = X u∈Nv,wu=m [v]u− [v+u]u+ xv+u, is a convergent series in an open neighborhood of some p∈Cn, both the neighborhood and p independent of m. We can take a generic perturbation ew∈Qnof wof the form ew=w+ǫeewith ee= (1,...,1) + ǫ′w′for ǫ > 0and ǫ′>0small enough and w′∈Qnis generic. Take any u∈Nv\ {0}and let us prove that wu ≥0. Since vis an exponent of HA(β)with respect to ewwe have by [26, (3.30)] that ewu > 0. Hence, since last inequality holds for ǫ > 0and ǫ′>0small enough we have that wu ≥0. Thus we have that wu ≥0for all u∈Nv. Notice that when wu > 0for all u∈Nv\ {0}then the set {u∈Nv, wu =m}is finite and hence fm(x)is clearly convergent. In general, {u∈Nv, wu =m}is not finite, but we will see that fm(x)is still convergent at some point p∈Cn. Since Nv= (−Bσk+NBσ)∩Znthe set {u∈ Nv, wu =m}is a finite union of shifted copies of the form N(i) = u(i) + (Pj /∈σ;wbj=0 Nbj)∩Zn where {bj:j /∈σ}is the set of columns of Bσand u(i)∈Nvsatisfies wu(i) = m. The series fm(x)is convergent if and only if all the series gi,m(x) = Pu∈N(i) [v]u− [v+u]u+ xv+uare convergent. Since βis very generic, nsupp(v) = ∅, and the convergence of each series gi,m is equivalent to the convergence of the series xv+u(i)Pu∈−u(i)+N(i) |u−|! |u+|!xu. Thus, it is enough to see that for any column uof Bσsuch that wu = 0 we have that |u|=|u+| − |u−| ≥ 0. Notice that wu = 0 implies 0<ewu =ǫ(|u|+ǫ′w′u)and so |u|+ǫ′w′u > 0. Hence, since this holds for ǫ′>0small enough, we have that |u| ≥ 0. We have proved that fis a formal solution of MA,w,wv(β)along Tand it is clear that f0(x) = xv+· · · 6= 0. From the expression of the gi,m and [10, Th. 3.11] any fm(x)is convergent at any point in {x∈Cn|06=Qi∈σxi,|xj|< R|xA−1 σaj σ|for j6∈ σand |wbj|= 0}for some R > 0. Q.E.D. Let ew∈Qnbe a generic perturbation of w∈Znas in the proof of Lemma 4. Lemma 5 If f(x, t) = Pm≥0fm(x)tγ+m∈tγOd X|T,p is a solution of MA,w,α(β)for some γ∈C, p∈T, with f0(x)6= 0, then: (a) tαf(x, t)is a solution of MA,w(β). 17 (b) For all m≥0,fm(x)is a holomorphic solution of MAw(β, α +γ+m), where this last module is the hypergeometric system associated with the matrix Awand the parameter (β, α +γ+m). (c) b(α+γ) = 0, where b(s)is the indicial polynomial of HA,w(β)along T. (d) If βis very generic then α+γ=wv for some exponent vof HA(β)with respect to ew. Proof. The proof of (a) and (b) are straightforward. Let us prove (c). By (a) and using [26, Theorem 2.5.5] we have that in(0,1)(tαf(x, t)) = f0(x)tα+γis a solution of inτ(HA,w(β)). Recall by definition of b(s)that hb(θt)i= inτ(HA,w(β)) ∩C[θt]where θt=t∂t. Thus, the differential operator b(θt)annihilates in(0,1)(tαf(x, t)) = f0(x)tα+γ. This implies, using for example [26, Lemma 1.3.2], that 0 = b(θt)(f0(x)tα+γ) = b(α+γ)f0(x)tα+γand this implies that b(α+γ) = 0. Let us prove (d). By (b) we have that f0(x)is a holomorphic solution of HAw(β, α +γ)and thus it can be written as a Nilsson series at the origin with respect to a vector eethat is a perturbation of e= (1,...,1) (see [26], [25], [8]) and, in particular, it makes sense to consider the initial form of f0(x)with respect to ee. On the other hand, using Lemma 3, we have that inw(IA) + hAθ −βi ⊆ inτ(HA,w(β)) annihilates f0(x). Since βis very generic, inw(IA) + hAθ −βi= in(−w,w)HA(β) [26, Th. 3.1.3]. This implies that f0(x)is a solution of in(−w,w)HA(β)and hence, inee(f0(x)) is a solution of in(−ew, ew)HA(β)for ew=w+ǫee. Thus, since βis very generic inee(f0(x)) = cxv for c∈Cand van exponent of HA(β)with respect to ew. Hence, using (b), we also have that wv =α+γ. Q.E.D. Remark 8 Although we assume in this paper that e A(w)is pointed it turns out that in this section this fact is only used in the proof of (d) in Lemma 5. However, let us notice that if e A(w)is not pointed and wis generic then inw(IA) = C[∂],inτ(HA,w(β)) = Dand so b(s) = 1. Thus, by (c) in Lemma 5 the modified system MA,w,α(β)does not have any solution in tγOd X|T,p for all γ∈C and p∈T. Remark 9 Since ew=w+ǫeewith ee= (1,...,1) + ǫ′w′for sufficiently small ǫ > 0and ǫ′>0 we have that inew(IA) = inw′(ine(inw(IA))) for e= (1,...,1). In particular, inew(IA)and inw(IA) have the same degree. Let us denote by dimC(M, F)pthe dimension of the space of F-solutions of a D-module Mat a point p. Theorem 5 Assumeβ∈Cdis very generic, w∈Znandα∈C. Then dimC(MA,w,α(β),Od X|T)p= 0if wv −α /∈Nfor all the exponents vof HA(β)with respect to ew. We also have that dimC(MA,w,α(β),X b(α+γ)=0 tγOd X|T)p= deg(inw(IA)) In particular, if wis generic we also have dimC(MA,w,wv−m(β),Od X|T)p= 1 for all generic p∈T,m∈Nand any exponent vof HA(β)with respect to w. 18 Proof. The first statement follows from Lemma 5, (d). The inequality dimC(MA,w,α(β),Pb(α+γ)=0 tγOd X|T)p≥deg(inw(IA)) follows from Lemma 4, Remark 7 and Remark 9. On the other hand, since the differential operators defining MA,w,α(β) belong to the Weyl Algebra we have that any solution f∈Pb(α+γ)=0 tγOd X|Tof MA,w,α(β) decomposes as a finite sum of solutions, each of them in a space tγOd X|T,p. Recall by the proof of Lemma 5 that any solution f∈tγOd X|T,p of MA,w,α(β)verifies that in(ew,0)(in(0,1)(f)) = cxvtwv−α for an exponent of HA(β)with respect to ew. This last fact, Remark 7, Remark 9 and a slightly modified version of [26, Proposition 2.5.7] prove that dimC(MA,w,α(β),Pb(α+γ)=0 tγOd X|T)p≤ deg(inw(IA)). Finally, the last statement follows from the second statement and from the fact that if βis very generic, w∈Znis generic and vand v′are two different exponents of HA(β)with respect to w, then w(v−v′)/∈Z. Q.E.D. Remark 10 Let ψv(x, t)be the series constructed in Lemma 4 and used in Theorem 5. If wis in the row span of Athen f(x, t) = tα−wvψvdoes not depend on tand thus it is a convergent series. If wis not in the row span of Athen f(x, t)is Gevrey along Twith index s=r+ 1 where r= max{− |u| wu :Nu⊆Nv, wu > 0} where |u|=Piui. On the other hand, as mentioned in the proof of Lemma 4 since βis very generic and vis an exponent of HA(β)with respect to ewthen vis associated with a simplex σ∈Φew,d−1 Aand there is a basis {bi:i /∈σ}of the kernel of Asuch that for all i /∈σ,(bi)j= 0 for all j /∈σ∪ {i}and (bi)i= 1. The set {bi:i /∈σ}is the set of columns of Bσ(if we reorder the variables so that σ={1, . . . , d}) and in this case we have that Nv= (−Bσk+NBσ)∩Zn. Thus, more explicitly, r= max{−|bi|/(wbi) : i /∈σ, wbi>0}where {bi:i /∈σ}is the set of columns of Bσ,|bi|= 1 − |A−1 σai|and wbi=wi−wσA−1 σai>0. The proof of this formula is technical and follows from standard estimates on Gamma functions similar to the ones used in [10] to compute the index of Gevrey solutions for hypergeometric systems. In particular, if wis a perturbation of (1,...,1) then ris close to −1and if Ais homogeneous then r= 0 because |u|= 0 for any u∈Nvand hence in both cases the series is convergent. 4.5 Gevrey solutions modulo convergent series By Theorem 5, if both α∈Cand β∈Cdare very generic then MA,w,α(β)does not have any nonzero solution in Od X|T,p for all p∈T. This is in contrast with the case of the irregularity of hypergeometric systems along coordinate hyperplanes, where for any slope s=r+ 1 of MA(β) along Y={xn= 0}and for very generic β∈Cdone can construct a formal solution φ∈ O d X|Y ,p of MA(β)along Y, such that φhas Gevrey index equal to the slope (see [10] and Theorem 3). However, by the comparison theorem for the slopes [22] and the perversity of the irregularity complex of a holonomic D-module along a smooth hypersurface [24], one knows that for each slope s=r+ 1 of MA,w,α(β)along Tat a generic p∈Tthere must exist a formal series φ∈ O d X|T,p with Gevrey index s=r+ 1 such that P(φ)is convergent at pfor all P∈HA,w,α(β). The purpose of this section is to describe Gevrey solutions modulo convergent series of the modified system MA,w,α(β)along Twhen α∈Cand β∈Cdare very generic. To this end, we use 19 the construction of the Gevrey solutions at infinity of the hypergeometric system Me A(w)(β, α −1) as performed in Theorem 4. Take X′=Cn+1 with coordinates (x1,...,xn, z)and z= 1/t so that X∩X′=Cn×C∗. Denote T′={t=∞} ={z= 0} ⊆ X′. We can consider for any γ∈Cthe C–linear map Υγ:tγOd X|T,p −→ t−1−γO\ X′|T′,p′ f=Pm≥0fm(x)tγ+m7−→ Υγ(f) = Pm≥0fm(x)[−γ−1]mt−1−γ−m where p= (p1,...,pn,0) ∈Tand p′= (p1,...,pn,∞)∈T′. Remark 11 Notice that Υγis an isomorphism if and only if γ /∈Z<0. In such a case we also have that Υγ(tγOd X|T(s−1)) = t−1−γO\ X′|T′(s)for all s. It is also clear that Υ0(Pm≥0fm(x)tk+m) = [−1]kΥk(Pm≥0fm(x)tk+m)for all k∈N. Theorem 6 Assume α∈Cand β∈Cdto be very generic. If s=r+ 1 >1is a slope of MA,w,α(β)along Tthen we can construct Pτvol(conv(0,eai:i∈τ)) Gevrey series that are linearly independent solutions of MA,w,α(β)modulo convergent series and whose Gevrey index is equal to s=r+ 1. Here τruns over all the facets of ∆Awsuch that −1 rean+1 ∈Hτand 0/∈τ. Moreover, the classes modulo Od X|T(<s)of these Gevrey series form a basis of the solution space of MA,w,α(β)in (Od X|T(s)/Od X|T(<s))pfor points p∈Tin a relatively open set of T. Proof. The existence of such facets τis given by Corollary 5. Since −1 rean+1 ∈Hτand −r= 2 −s′for s′=s+ 1 >2we have that s′>2is a slope of Me A(w)(β, α −1) along T′={t=∞}. Thus, by Theorem 4 we can construct Pτvol(conv(0,eai:i∈τ)Gevrey series along T′={t=∞} with index s′. Moreover, the classes in O\ X′|T′(s′)/O\ X′|T′(<s′)are linearly independent solutions of Me A(w)(β, α −1). More precisely, for any d-simplex σ⊆τthe series constructed are of the form φevfor ev= (v, −1−k)with Av =β,wv −1−k=α−1(i. e. wv −α=k∈N) and vi∈Nfor all i∈ {1,...,n} \ σ. Using Remark 11 we can take ψv(x, t)as the unique Gevrey series along Twith index s=r+1 verifying Υ0(ψv(x, t)) = φevfor ev= (v, −1−k). We conclude by Remark 11 that the Pτvol(conv(0,eai:i∈τ)series ψv(x, t)constructed are Gevrey series with index s=s′−1 = r+ 1 whose classes modulo Od X|T(<s)are linearly independent. Moreover, it can be checked that they are solutions of the modified system modulo OX|Tby using the fact that their images by the morphism Υ0are solutions of Me A(w)(β, α −1). Last statement follows from (7), [22] and [27]. Q.E.D. Example 3 Take A= (1 3 5),w= (0,1,1) and β, α ∈C. We have that e Ie A(w)=h∂2−t∂3 1, ∂3− t∂5 1iand HA,w,α(β) = De Ie A(w)+Dhx1∂1+ 3x2∂2+ 5x3∂3−β, x2∂2+x3∂3−t∂t−αi. Note here that e Ie A(w)is the binomial ideal generated by the operators in (3). The unique slope of MA,w,α(β) along Tis s=r+ 1 = 5 since −1 4ea4belongs to the line passing through ea1,ea3and σ={1,3}is a facet of ΦFLr e A(w)if and only if r≥4. 20 The volume of σis one and following the proofs of Theorem 6 and Theorem 4 we can take v= (β−5α, 0, α)which satisfies the conditions wv −α=k= 0 ∈N,v2= 0 ∈Nand Av =β. We get that the series ψv(x, t) = X m2,m2+m3≥0 [β−5α]3m2+5m3 [α+m3]m3m2!xβ−5α−3m2−5m3 1xm2 2xα+m3 3tm2+m3 is a Gevrey solution (modulo convergent series) of MA,w,α(β)along Twith index s=r+ 1 = 5. 5 Borel transformation and asymptotic expansion We assume that the Q-row span of the matrix Adoes not contain the vector (1,...,1), but the one of Awdoes, where Awis the matrix with columns eai,1≤i≤n(see Subsection 4.2). This case holds if and only if the weight vector wis in the image of ¯ ATwhere ¯ Ais the matrix a1··· an 1··· 1and it is not in the row span of A. In other words, the weight vector wlies in the intersection of the set of the secondary cones of ¯ A; see [29, Ch. 8]. Solutions of this case can be analyzed by utilizing the Borel transformation and the Laplace transformation. We review here some basics of the Borel summation method which we require in the following (see [3] for the details). Let us consider the formal expression f(t) = ∞ X ℓ=0 fℓtℓ+γ∈tγC[[t]] (11) where f06= 0 and γ∈C. Solutions constructed in Theorem 5 are of this form. If its coefficients satisfy fℓ≤CKℓΓ(1 + (ℓ+γ)/κ) (ℓ= 0,1,2,···)(12) with some positive constants C, K, κ, and ℜγ > −κ(in (19) this last condition will be relaxed), then the formal Borel transform (with index κ) defined by ˆ Bκ[f](τ) := ∞ X ℓ=0 fℓ Γ(1 + (ℓ+γ)/κ)τℓ+γ is the product of τγand a convergent power series at τ= 0. In addition to (12), if (i) the function ˆ Bκ[f]can be analytically continued to a sector S(θ, δ) := {reiθ′;θ′−θ< δ/2, r > 0} of infinite radius in a direction θ∈Rwith an opening angle δ > 0, and (ii) the analytic continuation of ˆ Bκ[f]satisfies the growth estimate ˆ Bκ[f](τ)≤c1exp c2τκ(13) in S(θ, δ)with some positive constants c1, c2>0, 21 then we say fis κ-summable in the direction θ, and define the κ-sum (or the Borel sum with index κ) of fby the Laplace transformation S[f](t) = Lθ κˆ Bκ[f](t) := Zeiθ·∞ 0 e−(τ/t)κˆ Bκ[f](τ)d(τ/t)κ,(14) where d(τ/t)κ=κτκ−1 tκdτ. Remark 12 Because of the growth condition (ii) of ˆ Bκ[f], the Laplace integral (14) converges if t satisfies ℜhτ tκi−c2|τ|κ>0.(15) Since ℜhτ tκi−c2|τ|κ=τ t κcos κ(θ−arg t)−c2|t|κ (note that arg τ=θ), the Laplace integral (14) converges in {t; cos κ(arg t−θ)] > c2|t|κ}.(16) The region (16) has infinitely many connected components. Here and in what follows we specify one of them by imposing |arg t−θ|< π/(2κ). Since we can vary arg τin (14) slightly, we conclude that S[f]defines a holomorphic function in [ |θ′−θ|<δ/2 {t; cos κ(arg t−θ′)] > c2|t|κ,|arg t−θ′|< π/(2κ)}. Therefore we can find ρ > 0and  > π/κ such that the S[f]is holomorphic in S(θ, , ρ) := S(θ, )∩ {t; 0 <|t|< ρ}(cf. [3, the first paragraph of §2.1]). Theorem 7 If fis κ-summable in a direction θ, then fis a Gevrey asymptotic expansion of its Borel sum S[f](t): For any closed subsector Sof S(θ, , ρ), there exists C′, K′>0for which the inequality t−γS[f](t)− N−1 X ℓ=0 fℓtℓ≤C′(K′)NtNΓ(1 + N/κ)(17) holds in Sfor N∈N. Proof. Here we give a sketch of the proof. See [3, Th. 1] for the details. It follows from the relation tℓ+γ=Zeiθ·∞ 0 e−(τ/t)κτℓ+γ Γ(1 + (ℓ+γ)/κ) κτκ−1 tκdτ (= Lθ κˆ Bκ[tℓ+γ]) 22 that the remainder of the expansion becomes S[f](t)− N−1 X ℓ=0 fℓtℓ+γ(18) =Zeiθ·∞ 0 e−(τ/t)κ(ˆ Bκ[f](τ)− N−1 X ℓ=0 fℓ Γ(1 + (ℓ+γ)/κ)τℓ+γ)κτκ−1 tκdτ. Since 1 τN+γ(ˆ Bκ[f](τ)− N−1 X ℓ=0 fℓ Γ(1 + (ℓ+γ)/κ)τℓ+γ) is holomorphic in some neighborhood of S(which includes the origin) and satisfies the growth condition (13) in Swith appropriate constants c1and c2, the righthand side of (18) can be estimated by the righthand side of (17) multiplied by |t|γ. Q.E.D. Inequality (17) is also known to be equivalent to conditions (i) and (ii) stated above ([3, Theorem 1 (p. 23)]). The Borel summation method may be also applied to the case when γ6∈ κZand ℓ+γ6∈ −κN>0for ℓ∈N.(19) We can define the Borel transform ˆ Bκ[f]in the same manner as before. In this last case, however, the Laplace integral (14) may not converge at τ= 0. Therefore we modify the definition of the Borel sum to S[f](t) = Lθ κˆ Bκ[f](t) := 1 1−e−2πiγ/κ ZΓκθ e−ζ/tκˆ Bκ[f](ζ1/κ)dζ tκ(20) with a path of integration Γκθ which runs from ∞along arg ζ=κθ −2πto some point near the origin, takes a 2πradian turn along a circle with the center at the origin, and goes back to infinity in the direction arg ζ=κθ. When condition (19) is satisfied and ℜγ > −κ, then (20) coincides with (14). The Borel sum (20) also satisfies the same properties as previously defined (14). We apply the Borel summation method to the Gevrey solution constructed in Theorem 5. Our main result of this section is Theorem 8 WeassumethattheQ-row span of the matrix Adoes notcontain the vector (1,1,...,1) but that the one of Awdoes. We also assume βto be very generic. Let ψ(x, t) = ∞ X ℓ=0 Cℓ(x)tℓ+γ(21) be one of the formal solutions of the modified hypergeometric system HA,w(β)constructed in Theorem 5 and r+ 1 be the Gevrey index of ψ(x, t)along T. We also assume rγ 6∈ Z. Then the formal solution ψ(x, t)is 1/r-summable (as a formal power series in t) in all but finitely many directions for each x∈Uwhere Uis a non-empty open set in the x-space Cn. Furthermore its Borel sum determines a solution of the modified hypergeometric system HA,w(β). 23 Remark 13 Under the assumption of Theorem 8, r=−|bi| w·bi (22) holds for any i6∈ σ, where {bi:i /∈σ}is the basis of Ker Agiven in Remark 10. Therefore if u∈Ker A∩Zn, then rw ·uis an integer. Remark 14 The condition that Awcontains (1,1,...,1) is assumed so that the Borel transformed series satisfies a regular holonomic system [18]. Hence, the growth condition (ii) of the Borel summability is satisfied because solutions of regular holonomic systems satisfy a polynomial growth condition. Without this assumption, things become more complicated. See also Section 6. Proof.(of Theorem 8). First of all, the open set Uin the theorem can be chosen as follows. There exist constants cij, ci, pi, m such that the series ϕB(x, ζ), which will be defined in the proof below, converges when (x1, . . . , xn, ζ)belongs to the non empty open set W=W′∩(∩n j=1(xj6= 0)) where W′is defined by the inequalities Pjcij log |xj|+cilog |ζ|< pi, for i= 1,...,m. Such constants exist because ϕB(x, ζ)is a hypergeometric series which satisfies a regular holonomic A-hypergeometric system ([14], [26, Section 2.5]); see also forthcoming Lemma 7. Since only non-negative powers of ζmodulo an exponent appear in ϕB, we may assume that ci>0for i= 1,...,m. We may choose a non empty domain U⊂Cnwith compact closure such that U× {ζ∈C|0<|ζ|< ǫ} ⊂ Wfor some ǫ > 0. To study the analytic properties of ˆ B1/r[ψ], it is convenient to use ϕ(x, z) := ψ(x, t)t=zr= ∞ X ℓ=0 Cℓ(x)zr(ℓ+γ).(23) Since t−γψ(x, t)is a formal power series in tw·bi,z−rγϕ(x, z)does not contain any fractional powers in z(cf. Remark 13). Then we have ˆ B1[ϕ](x, ζ) = ∞ X ℓ=0 Cℓ(x) Γ(1 + r(ℓ+γ))ζr(ℓ+γ)=ˆ B1/r[ψ](x, ζr).(24) In what followswe simply write ϕB(x, ζ)(resp., ψB(x, τ))insteadof ˆ B1[ϕ](x, ζ)(resp., ˆ B1/r[ψ](x, τ)). Lemma 6 Assume condition (19) holds for κ= 1/r. For the power series ϕgiven in (23), we have θζϕB=ˆ B1[θzϕ]and ∂ϕB ∂ζ =ˆ B1[z−1ϕ]. Proof. The first relation follows from ζ∂ϕB ∂ζ (x, ζ) = ∞ X ℓ=0 Cℓ(x) Γ(1 + r(ℓ+γ))r(ℓ+γ)ζr(ℓ+γ) =ˆ B1"∞ X ℓ=0 Cℓ(x)r(ℓ+γ)zr(ℓ+γ)#=ˆ B1z∂ϕ ∂z . 24 We also have ∂ϕB ∂ζ (x, ζ) = ∞ X ℓ=0 Cℓ(x) Γ(1 + r(ℓ+γ))r(ℓ+γ)ζr(ℓ+γ)−1 = ∞ X ℓ=0 Cℓ(x) Γ(r(ℓ+γ))ζr(ℓ+γ)−1 =ˆ B1"∞ X ℓ=0 Cℓ(x)zr(ℓ+γ)−1#=ˆ B1z−1ϕ. Q.E.D. Lemma 7 The formal power series ϕB(x, ζ)given in (24) formally satisfies the hypergeometric system HAB(βB), where AB=A0 w−1/r, βB=β 0. When the matrix ABcontains a rational entry, we regard the Z-module generated by the column vectors as the lattice to define the A-hypergeometric system. For example, when AB= 1 3 0 1 1 −1/2, βB= (β, 0), the lattice is Z×Z/2and the hypergeometric system is nothing but that for AB=1 3 0 2 2 −1and βB= (β, 0) for the lattice Z2. Proof. It follows from Lemma 6 and relations θjϕ=θjψ|t=zr,θzϕ(x, z) = r(θtψ)|t=zrthat n X j=1 aijθj−β!ϕB=ˆ B1" n X j=1 aijθj−β!ϕ# =ˆ B1  n X j=1 aijθj−β!ψt=zr = 0 and n X i=1 wiθi−1 rθζ!ϕB=ˆ B1" n X i=1 wiθi−1 rθz!ϕ# =ˆ B1" n X i=1 wiθi−θt!ψt=zr#= 0. Now we take vectors u= (u1,...,un+1)T, v = (v1,...,vn+1)T∈Nn+1 satisfying ABu= ABv. By its definition, we obtain 1 r(un+1 −vn+1) = n X i=1 wi(ui−vi)∈Z. 25 Proof. By [10, Theorem 3.11] we have that φv(x)is a Gevrey solution of MA(β)with index s=r+ 1 = maxi{|A−1 σai|}) along Y={xi= 0 : |A−1 σai|>1}at any point of Y∩ {x∈Cn: |xj|< R|xA−1 σaj σ|if j /∈σand |A−1 σaj|= 1} ∩ {xi6= 0 : i∈σ}, for some R > 0. It is clear from (29) that w∈Nnand wj= 0 for all j∈σ. Hence, for any exponent v+u in the series φv(x)the corresponding exponent of tin the series ψ(x, t)is −α+w(v+u) = −α+Pj /∈σwj(vj+uj)∈ −α+Nbecause vj∈Nfor all j /∈σand u∈Nv. We conclude the proof by using Remark 10. Q.E.D. In analogy with Section 5 we denote ψB(x, τ) = ˆ B1/r′[ψ](x, τ), which defines a holomorphic function at any point in Uσ,R for some R > 0by Proposition 4. We also denote ϕ(x, z) = ψ(x, zr′) and hence ϕB(x, ζ) = ˆ B1[ϕ](x, ζ) = ψB(x, ζr′)is convergent at points in the open set U′ σ,R ={(x, ζ)∈Cn×C:ζ=τ|det(Aσ)|,(x, τ)∈Uσ,R}(30) Moreover, the series ϕB(x, ζ)is a holomorphic solution of HAB(βB)(in the variables (x, ζ)), where AB=A0 w−κ, βB=β αwith κ=|det(Aσ)|. Remark 18 For wgiven by (29) the hypergeometric system HAB(βB), can have slopes along ζ=∞for all β, α ∈C(see Example 6). However, see Proposition 5 where we point out a property of the solution ϕB(x, ζ). Example 6 Let us consider the matrix A=2 0 1 3 0 1 1 2 , β∈C2,α∈C, the simplex σ={1,3}and w= (0,0,0,3) given by (29). Notice that det(Aσ) = 2 and hence AB=  2 0 1 3 0 0 1 1 2 0 0 0 0 3 −2  Using Corollary 2 we have that s= 1 + r= 1 + 1/3is a slope of MAB(βB)along ζ=∞. Lemma 10 Assume ΦF,d−1 Aη⊆ΦF,d−1 Awhere η={i:|A−1 σai| ≥ 1}. Let eσ⊆τ∈ΦF,d−1 Aηbe a simplex, ev∈Cn+1 a vector associated with eσ∪ {n+ 1}(i.e. ABev=βBand evi∈Nfor all i /∈eσ∪ {n+ 1}). The series φev(x, ζ)converges at points (x, ζ)∈U× {ζ:|ζ|> R′}and for arbitrarily small c2>0we can choose c1>0such that |φev(x, ζ)| ≤ c1exp(c2|ζ|). Remark 19 The previous condition on the (A, F)-umbrella holds for any d×nmatrix Awith d= 1 or n−1 = d. Proof. Notice that η∪ {n+ 1}is a facet of the (AB, F)-umbrella and eσ∪ {n+ 1}is a simplex of ABcontained in η∪ {n+ 1}. In particular we know that φevis convergent in certain open set. 32 Let us denote κ=|det(Aσ)|. If {bi:i /∈eσ∪ {n+ 1}} is the basis of ker(AB)associated with eσ∪ {n+ 1}then the coordinate sum of biis |bi|=0if i∈η\eσ −|A−1 eσai|+ 1 −1 κweσA−1 eσaiif i /∈η Let us denote by (bi)Athe vector given by the first nentries of bi. Since eσ∪{n+1} ⊆ η∪{n+1} ∈ ΦF,d−1 AB, we have that |bi|>0for all i /∈ηand that the series φevdefines a multivalued holomorphic function in the open set {(x, ζ) : |x(bi)Aζ(bi)n+1 |< R, i ∈η\eσ} ∩ {xj6= 0 : j∈eσ}for some R > 0. The fact that eσ⊆τ∈ΦF,d−1 Aη⊆ΦF,d−1 Aguarantees that −|A−1 eσai|+ 1 ≥0for all i∈ηand −|A−1 eσai|+ 1 >0for all i /∈η. Thus, if i∈η\eσthe last coordinate of biis (bi)n+1 =|bi| −|(bi)A|=−1 + |A−1 eσai| ≤ 0while if i /∈ηthe last coordinate of bican be positive. However, if (bi)n+1 >0for some i /∈η, we still have that |bi|>(bi)n+1 =−1 κweσA−1 eσai. In this case there exist K1, K2>0such that: X m≥0 (m!)−|bi||x(bi)Aζ(bi)n+1 |m≤K1exp(K2|x(bi)A|1/|bi||ζ|(bi)n+1/|bi|) where (bi)n+1/|bi|<1. On the other hand, if evi=kifor i /∈eσ∪ {n+ 1}, it can be shown by using standard estimates on Γ-functions (see e.g. [14, Proposition 1, Section 1.1], [25, Lemma 1] and [10, Lemma 3.8.]), that there exists C1, C2>0such that |φev(x, ζ)| ≤ C1|xA−1 eσβζ(−α+weσA−1 eσβ)/κ|X k+m∈Nn−d CPki+mi 2|xPi(ki+mi)(bi)AζP(ki+mi)(bi)n+1 | Qi/∈eσ∪{n+1}(ki+mi)!|bi|= =C1|xA−1 eσβζ(−α+weσA−1 eσβ)/κ|Y i/∈eσ∪{n+1} X ki+mi∈N (C2|x(bi)Aζ(bi)n+1 |)(ki+mi) (ki+mi)!|bi|! and for all i /∈eσ∪ {n+ 1}we have: X ki+mi∈N (C2|x(bi)Aζ(bi)n+1 |)(ki+mi) (ki+mi)!|bi|≤(K1exp(K2|C2x(bi)A|1/|bi||ζ|(bi)n+1/|bi|)if (bi)n+1 >0 1 1−|C2x(bi)Aζ(bi)n+1 |if (bi)n+1 ≤0 (31) Take U={x∈Cn:|x(bi)A|< Ri, i = 1,...,n}where Ri>0can be chosen arbitrarily large except when (bi)n+1 =|(bi)A|= 0 in which case we take Ri<1/C2. Then, since (bi)n+1/|bi|<1 if (bi)n+1 >0, we have the result for for arbitrarily small c2>0if we take c2>0and R′>0big enough. Q.E.D. Proposition 5 ϕB(x, ζ)has an analytic continuation to an open set of the form U×S(θ, δ), where Uis certain open set of Cnand S(θ, δ)is a sector with bisecting direction θand small enough opening δ > 0. Moreover, if ΦF,d−1 Aη⊆ΦF,d−1 Athen for arbitrarily small c2>0we can chose c1>0such that |ϕB(x, ζ)| ≤ c1exp(c2|ζ|)for (x, ζ)∈U×S(θ, δ). 33 Proof. To simplify the exposition we will first assume that αis very generic. Notice that η∪{n+1}is the set of (indices of) columns of ABbelonging to the hyperplane H= {|A−1 σy|− 1 |det(Aσ)|yd+1 = 1}(see Remark 17) and we denote by A′the submatrix of ABconsisting of these columns. Let qbe the cardinality of η. Recall that ϕB(x, ζ)defines a holomorphic function at each point of U′ σ,R (see Proposition 4 and (30)) and notice that wj= 0 for all j /∈η. We can write ϕB(x, ζ) = X m∈Nn−q ϕm xm η m! where ϕm=ϕm(xη, ζ)is a holomorphic solution of HA′(β−Pi/∈ηmiai, α), which is regular holonomic because all the columns of A′belong to the hyperplane H[18]. Let W⊂Cq+1 be the open set such that U′ σ,R =W×Cn−q(see (30)), so that for all m= (mi)i6∈η∈Nn−q,ϕmis holomorphic in W. Take Z={xi= 0 : i /∈η}and notice that we can identify Wwith a relative open subset of Z, i.e. with W× {0}=U′ σ,R ∩Z. Recall that the singular locus of a hypergeometric system does not depend on the parameter but only on the matrix (see [1] and [14]). In particular, since ϕmis convergent in Wfor all m, we can consider the analytic continuation of all the ϕmalong the same path starting at a point in Wand avoiding the singular locus of the hypergeometric system associated with A′. Let c= (ci)i∈η∈ Cqbe such that the complex line {xi=ci:i∈η} ∩ Z(with coordinate ζ) intersects Wat nonsingular points of HA′(β, α). Notice that this intersection is a relative open set in the complex line. Let Sing(c)be the set of points ζ0∈C\ {0}such that (xη, ζ) = (c, ζ0)is a singular point of HA′(β−Pi/∈ηmiai, α).Sing(c)is a finite set and thus Θ(c) = {arg(u) : u∈Sing(c)}is also finite. As we vary cin a small open set W′⊆Cq,Θ(c)is contained in a finite union of small intervals and we can take θsuch that for δ > 0small enough, (θ−δ/2, θ +δ/2) ∩Θ(x) = ∅for all x∈W′. Hence we can consider the analytic continuation of each ϕmto an open set containing W′×S(θ, δ). We have extended ϕBas a formal solution of MAB(β, α)along Z, which is convergent in some relative open set of Z. Thus, by the constructibility of the solutions of a holonomic system in the sheaf Od X|Z/OX|Z(see [24]) we have that the formal solution constructed is convergent at any point of W′×S(θ, δ)× {0}, thus ϕB(x, ζ)can be analytically continued to an open set containing W′×S(θ, δ)× {0}. Let us see that the analytic continuation of ϕB(x, ζ)satisfies a growth estimate near ζ= ∞. Each ϕmhas polynomial growth since it is a solution of the regular hypergeometric system MA′(β−Pi/∈ηmiai, α). Since α, β are very generic, each ϕmcan be written as a Nilsson series that converges in certain open set (see e.g. [26, Proposition 3.4.4]) which is a linear combination of series of the form φv(m)(xη, ζ)(for some set of exponents v(m)associated with simplices in certain regular triangulation of the matrix A′) with support Nv(m)given by integer vectors in ker(A′)with coordinates sum equal to zero. The open set where the Nilsson series converge depends on the regular triangulation of A′that the simplices belong to. We need to use a Nilsson series expression of ϕmthat converges in points (xη, ζ) = (c, ζ)with |ζ|> R for a sufficiently large R > 0. It is enough to consider a regular triangulation Tof Aηand take T′={eσ∪ {n+ 1}:eσ∈T}as the regular triangulation of A′. By properties of regular triangulations, there is one regular triangulation Tof Aηsuch that there exists cas above so that if |ζ|> R for a sufficiently large R > 0 then (c, ζ)belongs to the domain of convergence of the series φevfor any vector evassociated with 34 eσ∪ {n+ 1} ∈ T′. The series expression of ϕBvia substitution of each ϕmby its Nilsson series expansion is a formal Nilsson series (see e.g. [10, Lemma 6.15]). We know that this Nilsson series converges to ϕBat points in W′×S(θ, δ)× {0}close to ζ=∞. By Lemma 10 it verifies the desired growth estimate. Notice that the Nilsson series expansion of ϕBnear ζ=∞also provides an analytic continuation to points with |ζ|big enough and xvarying in certain open set of Cnthat contains (c, 0). We have considered analytic continuations along paths contained in Z. We can also extend ϕB by analytic continuation along paths from a point in W×Cn−qto a point near ζ=∞avoiding the singular locus of MAB(β, α). If the starting point of the path is close to (c, 0) the analytic continuation coincides with the Nilsson series close to ζ=∞and thus it satisfies the same growth estimate. Finally, the parameter βbeing very generic, the rank of MAB(βB)equals vol(AB)since the set of exceptional parameters has codimension at least 2 [23, Porism 9.5]. So, we can reduce the general case (when αis not necessarily generic) to the previous one following the ideas of the proof of [26, Theorem 3.5.1] (see also the proof of [10, Theorem 6.2. ]). Q.E.D. Remark 20 Using Proposition 5, the results in Section 5 also hold for was in (29) instead of w satisfying the assumption in Theorem 8 if we assume the additional condition ΦF,d−1 Aη⊆ΦF,d−1 Ato hold. In particular, we obtain an analogous version of Corollary 6. Let β∈Cnbe very generic and let φvbe a Gevrey solution of HA(β)of index s= 1+1/k > 1with respect to some coordinate subspace Z⊆Cn. Let σbe the simplex which vis associated with, so we have that φvis also a Gevrey series of index s= 1 + 1/k > 1with respect to Y={xi= 0 : |A−1 σai|>1} ⊃ Z. Note that if we take wassociated with σas in (29) then w(v+u)∈Nfor all u∈Nv. By Proposition 4 we have that tαψ(x, t) = φv(tw1x1,...,twnxn)is a Gevrey series along t= 0 of Gevrey index s′= 1 + 1/κ with κ=|det(Aσ)|. Let S[ψ](x, t)be the κ-sum of ψ(x, t)with respect to tin a direction θ /∈Θ(x)for xin certain open set Usmall enough with compact closure. For any closed subsector Sof S(θ, α, ρ)(see notations in Section 5) there exist constants C > 0, K > 0such that the inequality |tαS[ψ](c, t)−tαψN(c, t)| ≤ CKNΓ(1 + N/κ)|t|N holds for t∈S,c= (c1, . . . , cn)∈Uand any N∈N. Thus, considering parametric curve x(t) = (c1tw1,...,cntwn)then ψN(x(t),1) = X u∈Nv,w(v+u)≤N−1 [v]u− [v+u]u+ cv+utw(v+u)=tαψN(c, t) and S[ψ](x(t),1) = tαS[ψ](c, t). Note that xtends to the point x′∈Y, with x′ i=xiif |A−1 σai| ≤ 1, as ttends to 0. Theorem 9 Let β∈Cnbe very generic and let φvbe a Gevrey solution of HA(β)of order s= 1 + 1/k > 1with respect to a coordinate hyperplane Y={xi= 0}. Let σbe the simplex which vis associated with and take wassociated with σas well. If ΦF,d−1 Aη⊆ΦF,d−1 Athen for κ=|det(Aσ)|we have that S[ψ](x, 1) is a holomorphic solution of MA(β)and that for each (x1,...,xi−1, xi+1,...,xn)in certain open set of Cn−1,φv(x)is a Gevrey asymptotic expansion of order sof S[ψ](x, 1) with respect to xi= 0 in all but finitely many directions. 35 Proof.- Assume for simplicitythat the hyperplane is Y={xn= 0}and so σ⊆ {1,...,n−1}. We have that wi= 0 for i= 1,...,n−1and wn=|det(Aσ)|(s−1) >0where s=|A−1 σan|>1 is the Gevrey index of φvalong Y. By Remark 20 for x(t) = (c1,...,cn−1, cntwn)with t∈Sand c∈U, we have the inequality |S[ψ](x(t),1) −ψN(x(t),1)| ≤ CKNΓ(1 + N/κ)|t|N for all N≥0. Here we write ψN(x, 1) = Pwnm<N fm(x1,...,xn−1)xm n. For integers of the form N=wnM with M∈Nwe have that tN= (xn(t)/cn)Mand N/κ =M(s−1) = M/k where s= 1 + 1/k. Then we get the inequality |S[ψ](x(t),1) −ψN(x(t),1)| ≤ C(Kwn/|cn|)MΓ(1 + M/k)|xn(t)|M for all M≥0. This finishes the proof since we can assume by taken a smaller open set Uthat cn6= 0 and that |cn|> C′for some constant C′>0. Q.E.D. Example 7 Put A= (1 2 3),β∈Cand w= (0,0,1). The vector v= (0, β/2,0) is an exponent of the A-hypergeometric system HA(β)with respect to a perturbation of wand so the series ψ(x, t) = φv(x1, x2, tx3) = X m1,m3≥0,(m1+3m3)∈2Z [β/2](m1+3m3)/2 m1!m3!xm1 1x(β−m1−3m3)/2 2xm3 3tm3 is one of the series considered in the proof of Theorem 5 and it is a Gevrey solution of the modified system MA,w(β)along Twith order s=r+ 1 = 3/2. Notice that s=r+ 1 = 3/2is the Gevrey index of ψ(x, t)along Tif and only if β /∈2N(otherwise ψ(x, t)is a polynomial). Following Section 5 but with our vector w(which does not satisfy the assumptions in Section 5 but is of the form (29) for σ={2}) we consider the Borel transform of ψwith index κ= 1/r = 2: ψB(x, τ) = X m1,m3≥0,(m1+3m3)∈2Z [β/2](m1+3m3)/2 m1!m3!Γ(1 + m3/2)xm1 1x(β−m1−3m3)/2 2xm3 3τm3. This series defines a holomorphic function in {(x, τ)∈C4:|x3τ x3/2 2 |< ǫ, x1, x26= 0}for ǫ > 0 small enough and it has an analytic continuation with respect to τto certain sector S(θ, δ). Let us see that this analytic continuation has polynomial growth in τ. If ϕ(x, z) := ψ(x, t)|t=z1/2then its Borel transform (with index 1) ϕB(x, ζ) = X m1,m3≥0,(m1+3m3)∈2Z [β/2](m1+3m3)/2 m1!m3!Γ(1 + m3/2)xm1 1x(β−m1−3m3)/2 2xm3 3ζm3/2 is a solution of the hypergeometric system associated with ABand (β, 0) defined on C4with coordinates (x, ζ) = (x1, x2, x3, ζ). Notice that ϕBhas fractional powers in ζbut defines a multivalued holomorphic function in {(x, ζ)∈C4:|x2 3ζ x3 2|< ǫ, ζ 6= 0}for ǫ > 0small enough. It is clear that ϕB(x, ζ)is a linear combination of series φv(x, ζ)with v∈C4associated with the simplex {2,4}of AB(i.e., ABv=βB,vi∈Nfor i= 1,3). We have an analytic continuation of 36 ϕB(x, ζ)to a point in the open set {(x, ζ)∈C4:|x2 3ζ x3 2|> R}for R > 0big enough, which must be a linear combination of series φev(x, ζ)with ev∈C4associated with the simplex{3,4}(this simplex alone determines a regular triangulation of the of ABand the set of series φevwith evassociated with {3,4}generates the space of holomorphic solutions of MAB(βB)at any point in the open set {(x, ζ)∈C4:|x2 3ζ x3 2|> R}). We have that the columns of B{3,4}are b1= (1,0,−1/3,−1/6), b2= (0,1,−2/3,−1/3) and we notice that (b1)4=−1/6,(b2)4=−1/3<0. This implies that a series φev(x, ζ)with evassociated with {3,4}has polynomial growth as ζtends to infinity. Thus, the analytic continuation of ϕB(x, ζ)close to ζ=∞also does. As a consequence, the Borel sum of ψwith index 2given by S[ψ](x, t) = Lθ 2ˆ B2[ψ](x, t) = Zeiθ·∞ 0 e−(τ/t)2ψB(x, τ)d(τ/t)2 is a holomorphic solution of MA,w(e β)and S[ψ](x, 1) is a holomorphic solution of MA(β)which has an asymptotic expansion ψ(x, 1) that is Gevrey of order s= 3/2along x3= 0. Example 8 This example shows that the hypothesis in Proposition 5 on the umbrella is necessary and that the bound there is sharp. Take A=110ℓ 0 1 2 0  where ℓ > 1is a rational number. We can consider ℓ∈Qby changing the lattice Z2by the lattice (1 ℓZ)×Z. Then, the weight vector w= (0,0,0, ℓ−1) is associated with σ={1,2}by the formula (29) and κ=|det Aσ|= 1. Let α∈Cand β∈C2be very generic and let v∈C4be a vector associated with σso that the series φv(x)is a Gevrey solution of MA(β)along x4= 0 with Gevrey index s= 1 + r=ℓ > 1. Then for ψv(x, t) = t−αφv(x1, x2, x3, tℓ−1x4)the Borel transform ϕB(x, ζ)is convergent in the open set {(x, ζ) : |ζℓ−1|< ǫ|xℓ 1/x4|} for some ǫ > 0small enough. Moreover, it defines a holomorphic solution of MAB(βB)and then it is a linear combination of the set of series φv′(x, ζ)with v′∈C5associated with the simplex {1,2,5}. Its analytic continuation to points in the open set {(x, ζ) : |ζℓ−1|> R|xℓ 1/x4|} forsomesufficientlylarge R > 0isa linear combination of the set of series φevwith ev∈C5associated with the simplex {2,4,5}of AB. The column vectors of B{2,4,5}are b1= (1,0,0,−1/ℓ, (1 −ℓ)/ℓ)and b3= (0,−2,1,2/ℓ, 2(ℓ−1)/ℓ). Elements in the support Nevof φevare of the form m1b1+m3b3∈Z5with m1, m3∈N. Thus any of these series φev(x, ζ)is convergent in the open set {(x, ζ) : |ζℓ−1|> R|xℓ 1/x4|} for some sufficiently large R > 0and since the last coordinate of b3is 2(ℓ−1)/ℓ > 0there is a subseries of φev(the one with monomials (x, ζ)ev+m3b3,2m3(ℓ−1)/ℓ ∈N) such that the set of exponents of ζ in its monomials is contained in ev5+N. 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