Gevrey and formal Nilsson solutions of A-hypergeometric systems

Journal of Pure and Applied Algebra 224 (2020) 106350 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa Gevrey and formal Nilsson solutions of A-hypergeometric systems María-Cruz Fernández-Fernández 1 Departamento de Álgebra, Universidad de Sevilla, Av. Reina Mercedes S/N, 41012 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 5 August 2019 Received in revised form 17 January 2020 Available online 10 February 2020 Communicated by G.G. Smith MSC: 13N10; 33C70; 14M25; 32C38 Keywords: A–hypergeometric system D–module Gevrey series Formal Nilsson series Initial ideals We prove that the space of Gevrey solutions of an A–hypergeometric system along a coordinate subspace is contained in a space of formal Nilsson solutions. Moreover, under some additional condition, both spaces are equal. In the process we prove some other results about formal Nilsson solutions. © 2020 Elsevier B.V. All rights reserved. 0. Introduction We study formal solutions of A–hypergeometric systems, also known as GKZ–systems, as they were introduced by Gel’fand, Graev, Kapranov and Zelevinsky (see [7]and [8]). They are systems of linear partial differential equations associated with a pair (A, β) where Ais a full rank d ×nmatrix A =(aij) =(a1···an) with aj∈Zdfor all j=1, ..., nand β∈Cdis a vector of complex parameters. Recall that the toric ideal of Ais defined as IA:= ∂u+−∂u−|u∈Zn,Au=0⊆C[∂1,...,∂ n] where (u+)j=max{uj, 0}and (u−)j=max{−uj, 0}for j=1, ..., n. The A–hypergeometric system HA(β)is the left ideal of the Weyl algebra D=C[x1, ..., xn]∂1, ..., ∂ngenerated by IAand by the Euler operators Ei−βi:= n j=1 aijxj∂j−βifor i =1, ..., d. The A–hypergeometric D-module is nothing but E-mail address: [email protected]. 1MCFF was partially supported by MTM2016-75024-P, FQM-333 and FEDER. https://doi.org/10.1016/j.jpaa.2020.106350 0022-4049/© 2020 Elsevier B.V. All rights reserved. 2M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 the quotient MA(β) := D/HA(β). It is well known that MA(β)is a holonomic D-module, see [1]and [8]. In [1]and [8], it was also proved that the holonomic rank of MA(β), i.e. the dimension of the space of its holomorphic solutions at a nonsingular point, equals the normalized volume of A, denoted by vol(A) (see (1.1)), when βis generic. Moreover, MA(β)is regular holonomic if and only if IAis homogeneous (equivalently, (1, ..., 1) lies in the rowspan of the matrix A), see [10, Ch. II, 6.2, Thm.], [15, Thm. 2.4.11] and [19, Corollary 3.16]. Gevrey series solutions of a holonomic D–module Malong a variety Yare closely related with the irregularity sheaf of Malong Ydefined by Mebkhout [13]and with the so called slopes of Malong Y, see [11]. The slopes of MA(β)along a coordinate subspace Ywere computed in [19]. The spaces of Gevrey series solutions of MA(β)along Ywere described in [4](see also [6], [5]) for generic enough parameters β∈Cd. On the other hand, there is an algorithm that computes, for any regular holonomic left D–ideal Iand a generic vector w∈Rn, a set of canonical series solutions of Ithat belong to certain Nilsson ring. These series converge in a certain open set that depends on wand form a basis of holomorphic solutions of I, see [15, chapters 2.5 and 2.6]. In [3]the authors introduced a notion of formal Nilsson solutions of HA(β)in the direction of w, denoted by Nw(HA(β)), and they used it to generalize various results in [15]to the case when HA(β)is not necessarily regular. In the papers [4]and [3], some of the results assume βto be (very) generic, meaning that it lies outside a certain infinite (but locally finite) collection of affine hyperplanes. In particular, this condition is stronger than βbeing not rank jumping, a condition that only requires to avoid a concrete finite affine subspace arrangement of codimension at least two [12]. The set of rank jumping parameters is ε(A) := {β∈Cd| rank(MA(β)) >vol(A)}and it was computed in [12]in terms of the local cohomology modules of the toric ring SA=C[∂]/IA. In particular they proved that ε(A) =∅if and only if SAis Cohen–Macaulay. In this note we prove, for all β∈Cd, that the space of Gevrey series solutions of MA(β)along a coordinate subspace is contained in the space of formal Nilsson solutions of HA(β)in a certain direction, see Theorem 3.3. We also prove that under one additional condition both spaces coincide and that for β/∈ε(A) the dimension of this space is the normalized volume of certain submatrix of A, see Theorem 3.4. Moreover, in Section 2, we provide some additional results about formal Nilsson solutions of HA(β). Acknowledgments I am grateful to Christine Berkesch and Laura Felicia Matusevich for helpful conversations related to this work. I also thank two anonymous referees for useful comments and suggestions that improved the final version of this paper. 1. Preliminaries 1.1. Notations Let A =(a1···an)be a d ×nmatrix with columns aj∈Zdsuch that ZA := n j=1 Zaj=Zd. For a given subset τ⊆{1, ..., n}, set τ:= {1, ..., n} \τ. We shall identify τwith the set of columns of Aindexed by τand write Aτfor the submatrix of Awith column set τ. We denote by Δτthe convex hull in Rdof all the columns of Aτand the origin. We also denote pos(τ) := j∈τR≥0aj. We say that a subset σ⊆{1, ..., n}is a maximal simplex if Aσis an invertible matrix. We associate to a maximal simplex σan n ×(n −d)matrix Bσ, where its columns are indexed by σ, the j–th column of Bσ has σ–coordinates equal to −A−1 σaj, j–coordinate equal to one and the rest of coordinates equal to zero. In particular, the columns of Bσform a basis of the kernel of A. M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 3 For example, if σ={1, ..., d}then Bσ= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −A−1 σad+1 −A−1 σad+2 ··· −A−1 σan 10 0 01 0 . . ..... . . 00 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Recall that for any subset τ⊆{1, ..., n}, the normalized volume of Aτ(with respect to the lattice Zd) is given by: vol(Aτ):=d!volRd(Δτ) (1.1) where volRd(Δτ) denotes the Euclidean volume of Δτ⊆Rd. We also denote vol(τ) := vol(Aτ). If σ⊆ {1, ..., n}is a maximal simplex, then vol(σ) =[Zd:ZAσ] =| det(Aσ)|. 1.2. Regular triangulations A vector w∈Rndefines an abstract polyhedral complex Twwith vertex set contained in {1, ..., n}as follows: τ∈Twiff there exists a vector c ∈Rdsuch that c,a j=wjfor all j∈τ, (1.2) c,a j<w jfor all j/∈τ. (1.3) Such a polyhedral complex is called a regular subdivision of Aif it satisfies pos(A) =∪τ∈Twpos(τ). This happens for example if w∈Rn >0or if Ais pointed, i.e. the intersection of Rn >0with the rowspan of Ais nonempty. An element τ∈Twis called a facet of Twif the rank of Aτis d. Any regular subdivision is determined by its facets and from now on we will write only τ∈Twwhen τis a facet of Tw. We say that a regular subdivision Twof Ais a regular triangulation of Aif all its facets are simplices. An important case of regular subdivision of Ais the following. If wj=1for all j=1, ..., n, a facet of Tw is the same as a facet of ΔAnot containing the origin. This particular regular subdivision of Ais denoted by ΓA. Notice that for a maximal simplex σ, it is straightforward from (1.2)and (1.3)that σ∈Tw⇐⇒ wBσ>0.(1.4) If Tis any regular triangulation of Athen the set C(T):={w∈Rn|T=Tw}={w∈Rn|wBσ>0,∀σ∈T}(1.5) is an open nonempty convex rational polyhedral cone. The closures of these cones and their faces form the so called secondary fan of A, introduced and studied by Gel’fand, Kapranov and Zelevinsky [9, Chapter 7]. When Ais pointed, it is easy to see that the secondary fan is a complete fan, i.e. its support is Rn. In general, it is not necessarily complete but its support contains the orthant Rn ≥0. 4M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 1.3. The A–hypergeometric fan A vector w∈Rndefines a partial order on the monomials of the Weyl Algebra D(and also on the monomials in C[∂1, ..., ∂n]) by defining the (−w, w)–weight of xα∂γ∈Das the real value w, γ−α. The initial form of an element P=α,γ∈Nncα,γ xα∂γ∈Dwith respect to (−w, w), denoted by in(−w,w)(P), is the sum of the terms cα,γxα∂γ, with cα,γ =0, whose (−w, w)–weight is maximum. If I is a left D–ideal, its initial ideal with respect to (−w, w)is defined as in(−w,w)(I):=in(−w,w)(P)|P∈I, P =0. Remark 1.1. When Ais pointed, there is a vector w∈Rn >0in the rowspan of Aand we have that inw(IA) = IAand in(−w,w)(HA(β)) =HA(β). Then, for any w∈Rnwe have that w := w+w ∈Rn >0, inw(IA) = inw(inw(IA)) =in w (IA)and in(−w,w)(HA(β)) =in (−w,w)(in(−w,w)(HA(β))) =in (−w ,w )(HA(β)) for  >0small enough, see [15, Lemma 2.1.6]. The Gröbner fan of IA, see [18, p. 13] (resp. the small Gröbner fan of HA(β), see [15, p. 60]) is a rational polyhedral fan in Rnwhose cones Csatisfy that inw(IA) =in w(IA)(resp. in(−w,w)(HA(β)) = in(−w,w)(HA(β))) for all w, w∈˚ C, where ˚ Cdenotes the relative interior of C. By Remark 1.1, these two fans are also complete fans when Ais pointed. Definition 1.2. The A–hypergeometric fan (at β) is the coarsest rational polyhedral fan in Rnthat refines both the Gröbner fan of IAand the small Gröbner fan of HA(β). Remark 1.3. We notice that this fan is a refinement of the hypergeometric fan defined in [15, Section 3.3] when IAis homogeneous and βis generic. By [18, Proposition 8.15] and [2, Corollary 4.4] the A– hypergeometric fan is a refinement of the secondary fan of A. 1.4. Γ–series Let us denote kerZ(A) := {u ∈Zn| Au =0}. Following [8], for any vector v∈Cnsuch that Av =β, we consider the Γ–series ϕv:=  u∈kerZ(A) xv+u Γ(v+u+1), where Γ(v+u +1) =n j=1 Γ(vj+uj+1)and Γis the Euler Gamma function. These series are formally annihilated by HA(β). Moreover, when IAis homogeneous and β∈Cdis generic, a basis of convergent Γ–series solutions of MA(β)can be constructed by using any regular triangulation of A, see [8]. These Γ–series are handled in [15, Section 3.4] in the following way: φv:=  u∈Nv [v]u− [v+u]u+ xv+u(1.6) where Nv={u ∈kerZ(A)| ∀j=1, ..., n, vj+uj∈Z<0iff vj∈Z<0}and [v]u= n  j=1 vj(vj−1) ···(vj−uj+1). Set nsupp(v) := {j∈{1, ..., n}| vj∈Z<0}for any v∈Cn. M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 5 The series φvis annihilated by HA(β)if and only if vhas minimal negative support, i.e. there is no u ∈kerZ(A)such that nsupp(v+u) nsupp(v), see [15, Proposition 3.4.13] whose proof works as well when IAis not homogeneous. Remark 1.4. It is easy to check that Γ(v+1)ϕv=φvwhen v∈(C\Z<0)n. Notice that ϕv=ϕv+ufor any u ∈kerZ(A). Thus, for u ∈Nvthere is a nonzero scalar c ∈Csuch that φv=c ·φv+u. 1.5. Gevrey series solutions of A-hypergeometric D-modules In this section we introduce the notion of Gevrey series and we recall some notations and results from [4]. Let us denote, for a subset τ⊆{1, ..., n}, Yτ:= {xj=0| j∈τ}and xτfor the set of variables xjwith j∈τ. We denote by OXthe sheaf of holomorphic functions on X=Cnand by O X|Yτthe sheaf of formal series along Yτ. A germ of O X|Yτat p ∈Yτcan be written as f= α∈Nτ fα(xτ)xα τ∈O  X|Yτ,p ⊆C{xτ−pτ}[[xτ]] where fα(xτ) ∈O Yτ(U)for certain nonempty relatively open subset U⊆Yτ, p ∈U. A formal series f=α∈Nτfα(xτ)xα τ∈O  X|Yτ,p is said to be Gevrey of order s ∈Ralong Yτat p ∈Yτif the series  α∈Nτ fα(xτ) (j∈ταj!)s−1xα τ is convergent at p. Since MA(β)is a holonomic D–module, any of its formal solutions along Yτis Gevrey of some order. We denote by HomD(MA(β), O X|Yτ,p)the space of all Gevrey solutions of MA(β)along Yτat p ∈Yτ. Given a maximal simplex σand a vector k=(ki)i/∈σ∈Nσwe denote by vk σ∈Cnthe vector with σ–coordinates equal to A−1 σ(β−Aσk)and σ–coordinates equal to k. Let Ωσ⊆Nσbe a set of representatives for the different classes with respect to the following equivalence relation in Nσ: we say that k∼kif and only if Aσk−Aσk∈ZAσ. Thus, Ωσis a set of cardinality vol(σ) =[Zd:ZAσ]. When βis generic, the space of Gevrey solutions of MA(β)along Yτis explicitly described in [4]. Theorem 1.5. [4, Theorem 6.7 and Remark 6.8] If T(τ)is a regular triangulation of Aτthat refines ΓAτ and β∈Cdis generic enough, the set {φvk σ:σ∈T(τ), k∈Ωσ}is a basis of the space of Gevrey series solutions of MA(β)along Yτat any point pof a certain nonempty relatively open set WT(τ)⊆Yτ. In particular, dimC(HomD(MA(β), O X|Yτ,p)) =vol(τ). For more precise statements, including the Gevrey order of these series and its relation with the so called slopes of MA(β)along Yτ, see [4]. The slopes of MA(β)along Yτwere described in [19]. Notice that if τ=Athen Yτ=Cn, O X|Yτ=OX, and Theorem 1.5 gives a basis of holomorphic functions of MA(β)at any point of WT(τ)when β∈Cdis generic. Such a basis was first described in [14](see also [8]when IAis homogeneous). The following result is the first part of [4, Theorem 6.2]. Theorem 1.6. If pis a generic point of Yτthen, for all β∈Cd, dimC(HomD(MA(β),O X|Yτ,p)) ≥vol(τ). 6M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 1.6. Formal Nilsson solutions of A-hypergeometric D-modules We recall here some definitions and results from [3], see also [15]when IAis homogeneous. In the former paper the authors write the following results in terms of a regular triangulation of the matrix ρ(A) := (a0a1···an), that is constructed from Aby adding a first column of zeroes and then a first row of ones. It follows from the definition of regular subdivision (see (1.2)and (1.3)) that for a subset σ⊆{1, ..., n}, we have that σ∈Twif and only if {0} ∪σ∈T(0,w). Given a cone C⊆Rn, the dual cone of C, denoted by C∗, is a closed cone consisting of vectors u ∈Rn such that w, u ≥0for all w∈Cand all u ∈C ∗. If Cis full dimensional, then the cone C∗is strongly convex (i.e. it doesn’t contain non trivial linear subspaces) and w, u >0for all w∈˚ Cand all nonzero u ∈C ∗ We say that w∈Rn >0is a weight vector (for HA(β)) if it belongs to the interior of a full dimensional cone of the A–hypergeometric fan. For a weight vector w∈Rnwe denote by Cwthe interior of the (full dimensional) cone in the A– hypergeometric fan such that w∈C w. Notice that Cw⊆C(Tw). Definition 1.7. [3, Definition 2.6] Let wbe a weight vector for HA(β). Write log(x) =(logx1, ..., log xn). A basic Nilsson solution of HA(β)in the direction of wis a series of the form φ=xv u∈C xupu(log(x)),(1.7) where v∈Cn, that satisfies i) φis annihilated by the partial differential operators of HA(β); ii) Cis contained in kerZ(A) ∩C ∗for some strongly convex open cone C⊆C wsuch that w∈C; iii) the puare nonzero polynomials and there exists K∈Zsuch that deg(pu) ≤Kfor all u ∈C; iv) 0 ∈C. The set supp(φ) ={v+u| u ∈C}is called the support of φ. The C–linear span of all basic Nilsson solutions of HA(β)in the direction of wis called the space of formal Nilsson solutions of HA(β)in the direction of wand it is denoted by Nw(HA(β)). We define the w–weight of a term xvp(log(x)), where v∈Cnand pis a polynomial, as the real value Re(w, v). For a series φconsisting in a (possibly infinite) sum of terms, we say that it has an initial form if there exists the minimum for the set of w–weights of all its nonzero terms. In this case, its initial form in the direction of w, denoted by inw(φ), consists in the sum of all the terms of φwith minimum w–weight. Remark 1.8. Notice that if a series φas in (1.7)satisfies all conditions in Definition 1.7 we have inw(φ) = xvp0(log(x)). Proposition 1.9. [3, Proposition 2.11] Let w∈Rnbe a weight vector for HA(β), then dimC(Nw(HA(β))) ≤ rank(in(−w,w)(HA(β))). We recall that a vector v∈Cnis called an exponent of HA(β)with respect to wif xvis a solution of in(−w,w)(HA(β)). Theorem 1.10. [3, Theorem 4.8] If βis generic and wis a weight vector for HA(β)then the set {φv|vis an exponent of HA(β)with respect to w} M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 7 is a basis of Nw(HA(β)), where φvis defined in (1.6). Theorem 1.11. [3, Corollaries 4.9 and 4.11] If βis generic and wis a weight vector for HA(β)then dimC(Nw(HA(β)) = rank(in(−w,w)(HA(β))) = deg(inw(IA)) =  σ∈Tw vol(σ).(1.8) Let w∈Rn >0be a weight vector for HA(β). We say that wis a perturbation of a vector w0∈Rnif there exists a full dimensional cone Cof the A–hypergeometric fan such that w0∈Cand w∈˚ C. We notice that a weight vector wis a perturbation of (1, ..., 1) if and only if the regular triangulation Twis a refinement of ΓA. Theorem 1.12. [3, Theorem 6.4] Assume that Ais pointed. If wis a perturbation of (1, ..., 1) then, for all β∈Cd, dimC(Nw(HA(β))) = rank(in(−w,w)(HA(β)) = rank(MA(β)).(1.9) More precisely, Nw(HA(β)) is the space of convergent series solutions of MA(β)at any point in a certain nonempty open set Uw⊆Cn. 2. Some remarks on formal Nilsson solutions of HA(β) In this section we provide some additional results about Nw(HA(β)). Lemma 2.1. For any β∈Cdand any weight vector w, dimC(Nw(HA(β)) ≥ σ∈Tw vol(σ).(2.1) Proof. By Theorem 1.11 equality holds in (2.1)when βis generic and, by Theorem 1.10, there is a basis of Nw(HA(β)) that consists of the set of series φv, see (1.6), for vvarying in the set of exponents of HA(β) with respect to w. In this situation, when βis not generic, we can apply the same procedure as in the proof of [15, Theorem 3.5.1] and obtain a set of linearly independent formal Nilsson solutions of HA(β)in the direction of w. The cardinality of this set is the rightmost quantity in (2.1). 2 Corollary 2.2. If wis a weight vector, then (1.8)holds for any β∈Cd\ε(A). Proof. By [19, Theorem 4.28] and [2, Lemma 3.1], rank(in(−w,w)(HA(β))) is constant for β∈Cd\ε(A) and hence the second equality in (1.8)holds in this case too. The first equality in (1.8)now follows from Lemma 2.1 and Proposition 1.9.2 The following result states that the basis given in Theorem 1.10 only depends, up to multiplication of their elements by nonzero scalars, on the regular triangulation Twand not on the cone Cw⊆C(Tw). Proposition 2.3. If β∈Cdis generic and wis a weight vector, then the set Bw(β):={φvk σ:σ∈Tw,k∈Ωσ} is a basis of Nw(HA(β)). 8M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 Proof. By the assumption on β, the difference between two vectors in the set {vk σ| σ∈Tw, k∈Ωσ} is not an integer vector. Thus, the series in Bw(β)have pairwise disjoint supports, hence they are linearly independent. Moreover, nsupp(vk σ) =∅, which implies that φvk σis annihilated by HA(β), see [15, Proposition 3.4.13]. On the other hand, the support of φvk σis the set supp(φvk σ)={vk σ+(Bσm)t|m∈Zσ,B σm∈Znand k+m∈Nσ}. Notice that vk σ+(Bσm)t=v0 σ+(Bσ(k+m))twhere k+m∈Nn. Thus, since wBσ>0for any σ∈Tw, we have that Re(w, v) ≥Re(w, v0 σ)for all v∈supp(φvk σ). It follows that there exists the initial form inw(φvk σ)and that it consists in a finite sum of terms. By the proof of [15, Theorem 2.5.5] we also have that inw(φvk σ)is a solution of in(−w,w)(HA(β)), but a basis of its solutions is given by the set of monomials xv for vvarying in the set of exponents of HA(β)(see Theorems 1.10 and 1.11). It follows that there exists an exponent vk σof HA(β)with respect to wsuch that vk σ∈supp(φvk σ), hence φvk σ=cσ,k·φvk σfor some nonzero scalar cσ,k∈C, see Remark 1.4. This implies that Bw(β)is a basis of Nw(HA(β)) in this case, by Theorem 1.10.2 Corollary 2.4. If w, ware weight vectors for HA(β), then we have the following: i) If βis generic, Nw(HA(β)) =Nw(HA(β)) if and only if Tw=Tw. ii) For all β∈Cd\ε(A), Nw(HA(β)) =Nw(HA(β)) if Cw=Cw. iii) For all β∈Cd\ε(A), the cone Cin Definition 1.7 can be chosen to be Cwfor any basic Nilsson solution of HA(β)in the direction of w. Proof. Proposition 2.3 directly implies i). Let us prove ii). We can take a basis B={φ1, ..., φr}of Nw(HA(β)) so that each φi=xv(i)u∈Cip(i) u(log(x)) is a basic Nilsson solution of HA(β)in the direction of w. Thus, for i =1, ..., r, there exists a strongly convex open cone Cias in condition ii) of Definition 1.7, i.e. w∈C i⊆C wand Ci⊆C ∗ i∩kerZ(A). Then the strongly convex open cone C:= ∩r k=1Ck⊆C wsatisfies that condition for all i =1, ..., r, since C⊆C iimplies C∗ i⊆C ∗. It follows that for all w∈C, the series φi are also basic Nilsson solutions in the direction of wand, by Remark 1.8, inw(φi) =xv(i)p(i) 0(log(x)). This implies that Nw(HA(β)) ⊆N w(HA(β)), for all w∈C. But this last inclusion must be an equality since both spaces have the same dimension, see Corollary 2.2. It follows that the space Nw(HA(β)), its subset of basic Nilsson solutions φiand their initial forms inw(φi)are locally constant with respect to the weight vector w, hence they are constant in the whole open cone Cwof the A–hypergeometric fan at β. This proves ii). For iii), notice that, the fact that inw(φi) =xv(i)p(i) 0(log(x)) for all w∈C wimplies that w, u ≥0for all u ∈Ciand all w∈C w. Thus, Ci⊆C ∗ wfor all i =1, ..., r, which proves the result. 2 Remark 2.5. If βis generic and w, ware weight vectors such that Tw=Tw, it may happen that inw(φ) = inw(φ)for some series φ ∈B w(β) =Bw(β)(in which case in(−w,w)(HA(β)) =in (−w,w)(HA(β))). For example, let us consider the matrix A=1111 0123  and a generic β∈C3. The maximal simplex σ={1, 4}defines a regular triangulation of Ainduced by any of the weight vectors w(1) =(1, 2, 5, 1) and w(2) =(1, 5, 2, 1). We can choose Ωσso that the series φvwith v=(β1−(β2+2)/3, 1, 0, (β2−1)/3), see (1.6), belongs to Bw(1) (β) =Bw(2) (β). Notice that Nv={u=(−(2m2+m3)/3,m 2,m 3,−(m2+2m3)/3) ∈Z4|m2≥−1,m 3≥0}. M.-C. Fernández-Fernández / Journal of Pure and Applied Algebra 224 (2020) 106350 9 Thus, inw(1) (φv) =xv=in w(2) (φv)since v+(0, −1, 2, −1) ∈supp(φv) =v+Nvhas smaller w(2)-weight than v. The following result improves [3, Corollary 6.9]. Corollary 2.6. If wis a weight vector, we have, for all β∈Cd, that dimC({φ∈N w(HA(β))|φis convergent})≥ σ∈T0 w vol(σ) (2.2) where T0 w={σ∈Tw| σ⊆ηfor some η∈ΓA}. Moreover, equality holds in (2.2)if βis generic. Proof. Assume first that β∈Cdis generic. Thus, Bw(β)is a basis of Nw(HA(β)) by Proposition 2.3. On the other hand, it follows from [4, Theorem 3.11] and the definition of Bσ, that φvk σis convergent if and only if (1, ..., 1)Bσ≥0, that holds if and only if σis contained in a facet of ΓA. Thus, B0 w(β) := {φvk σ:σ∈ T0 w, k∈Ωσ}is a linearly independent set of convergent formal Nilsson solutions of HA(β)in the direction of w. The equality in (2.2)for generic βfollows from the fact that the series in Bw(β) \B0 w(β)are all divergent and have pairwise disjoint supports, so no linear combination of them can be convergent. If β∈Cdis not assumed to be generic, we can consider a generic parameter βand apply to the set B0 w(β+β), with  ∈C such that ||is small enough, the same method as in [15, Theorem 3.5.1] to get the desired lower bound in (2.2). 2 Lemma 2.7. [16, Lemma 5.3] Let p(y) ∈C[y], v∈Cnand ν∈Nn, then ∂ν[xvp(log(x))] = xv−ν 0≤ν≤ν λν∂ν−ν[p](log(x)) where the sum is over ν∈Nnsuch that ν j≤νjfor all j, and λν∈C. In particular, λν=[v]ν= n  j=1 vj(vj−1) ···(vj−νj+1). The following lemma guarantees that in the third condition of Definition 1.7 we can assume that the constant Kis independent of β. Lemma 2.8. Let wbe a weight vector. Then for any basic Nilsson solution φof HA(β)in the direction of w as in (1.7)we have that deg(pu)≤(n+ 1)(22(d+1)vol(A)−1) for all u ∈C. Proof. Assume first that IAis homogeneous. In this case MAτ(β−Aτα)is regular holonomic [10]and deg(pu) ≤n(22dvol(A) −1) by [15, Theorem 2.5.14 and Corollary 4.1.2]. If IAis not homogeneous, we can consider the (d +1) ×(n +1) matrix ρ(A)as defined at the beginning of Subsection 1.6. Notice that φis also a basic Nilsson solution of HA(β)in the direction of wfor all w∈C, where Cis an open cone as in condition ii) in Definition 1.7. In particular, we can assume without loss of genericity that wis generic. This implies that (0, w) +λ(1, ..., 1) ∈Rn+1 is generic if λ >0is generic. Since (1, ..., 1) belongs to the rowspan of ρ(A), we have that (0, w)is a weight vector for Hρ(A)(β0, β), where β0∈C(see also [3, Remark 2.5]).