Irregular hypergeometric systems associated with a singular monomial curve

IRREGULAR HYPERGEOMETRIC SYSTEMS ASSOCIATED WITH A SINGULAR MONOMIAL CURVE MAR´ IA ISABEL HARTILLO-HERMOSO Abstract. In this paper we study irregular hypergeometric systems defined by one row. Specifically, we calculate slopes of such systems. In the case of reduced semigroups, we generalize the case studied by Castro and Takayama. In all the cases we find that there always exists a slope with respect to a hyperplane of this system. Only in the case of an irregular system defined by a1×2 integer matrix we might need a change of coordinates to study slopes at infinity. In the other cases slopes are always at the origin, defined with respect to a hyperplane. We also compute all the L-characteristic varieties of the system, so we have a section of the Gr¨obner fan of the module defined by the hypergeometric system. 1. Introduction Algebraic analysis or D-module theory studies systems of linear partial differential equations from the point of view of algebra and it generalizes the classical theory of ordinary differential equations with holomorphic coefficients in one complex variable x. A basic concept in the theory of ordinary linear differential equations with holomorphic coefficients is that of regular singular point. Fuch’s theorem [8] gives a combinatorial method to determine when a singular point is regular or irregular. Given an ordinary linear differential equation am(x)∂m+am−1(x)∂m−1+···+a0(x),with am(x)=0, we associate to it a combinatorial object, the Newton polygon, defined as convex hull   m  j=0 (j, j −val(aj(x))) + (−N)2 , where val(aj(x)) is the order of the holomorphic function aj(x)atx=0. Fuch’s theorem assures that the ordinary differential equation has a regular singular point at the origin if and only if its Newton polygon is a quadrant. The notion of slope arises from this theorem: if the Newton polygon has a slope, then the origin is an irregular singular point. We say that the equation is regular if it has no irregular singular points. Received by the editors July 15, 2003 and, in revised form, January 21, 2004. 2000 Mathematics Subject Classification. Primary 32C38; Secondary 13P10, 13N10, 33C80, 34M35. Key words and phrases. D-module, slopes, hypergeometric systems, Gr¨obner basis. This work was partially supported by FQM-813, FQM-333, DGESIC BFM2001-3164 and HF2000-0044. 4633 4634 MAR´ IA ISABEL HARTILLO-HERMOSO To generalize the theory of linear partial differential equations we introduce the Weyl algebra: An=Cx1,...,x n,∂ 1,...,∂ n, that is, the ring of differential operators with polynomial coefficients in nvariables. This ring is not commutative and the above elements verify the following relations: [xi,x j]=0,[∂i,∂ j] = 0, and [∂i,x j]=δi,j,whereδi,j is the Kronecker delta. We also need to introduce the ring Dn: Dn=C{x1,...,x n}∂1,...,∂ n, with the same relations between the generators. The first differential equation we considered can be seen as an operator in this ring. Every system of linear partial differential equations can be represented as aleftidealinDn,orinAnif the coefficients are polynomial. This can be done because a solution of such a system is also a solution for every element which comes from a combination (as left ideal) of those operators. In general we denote by Mthe left ideal generated by the elements of M.To simplify notation we usually note A,Dand xand ∂instead of (x1,...,x n)and (∂1,...,∂ n), if there is no confusion. The generalization of irregularity in several variables is not obvious. The equivalent notion is that of irregular sheaf, a concept due to Mebkhout (see [13]). Given a system, we consider the left ideal associated and the module defined by this ideal; if the irregular sheaf with respect to every hypersurface associated to this module is zero, we say that the system is regular. There also exists a generalization of the concept of slope; it is due to Laurent [9] and it describes an invariant associated to a D-module with respect to a hypersurface in one of its points. As in the case of one variable, those concepts are related, for holonomic modules. The slopes of a module describe the jumps in the Gevrey filtration of that sheaf [10]. We have that a holonomic D-module in several variables is regular if and only if it has no slopes for all the hypersurfaces ([10]). The slopes are used very often to determine if a module is irregular. There exists an algorithm, due to Assi, Castro and Granger [2], which calculates, using non-commutative Gr¨obner basis, the slopes of a D-module with respect to a hypersurface. In our work, we shall not use this algorithm because, although it is very useful in particular cases, it seems untreatable (so many Gr¨obner basis calculations) in general ones. The aim of our study are the hypergeometric systems defined by one row. The hypergeometric systems, in general, are systems of partial differential equations with polynomial coefficients, so it defines a left ideal in the Weyl algebra. Given a d×ninteger matrix Awith rank(A)=dand a complex vector β∈Cd,letHA(β) denote the left ideal generated by ∂u−∂v,u,v∈Nnsuch that Aut=Avt,(1.1) n  j=1 aijxj∂j−βi,i=1,...,d.(1.2) The operators defined in (1.1) are called toric operators, and the ideal defined by them in the commutative subring C[∂1,...,∂ n] is called the toric ideal and it is usually denoted by IA. We shall also use the notation IAto denote the extension of IRREGULAR HYPERGEOMETRIC SYSTEMS 4635 that ideal in the Weyl algebra or in D. On the other hand, for the operators defined in (1.2), that are usually called homogeneity operators, we shall use the notation θi=xi∂iand we represent those operators as the vector of operators Aθt−β. The ideal HA(β)⊂A nis called the A-hypergeometric system with parameter β. These systems were introduced by Gel’fand, Kapranov and Zelevinski [4]. We usually denote by HA(β) the module A/HA(β). The module defined by HA(β)is holonomic; see [4] and [1] for the general case. A result by Hotta [7] assures that if the toric ideal IAis homogeneous with respect to the usual grading, the hypergeometric system is regular. This condition, in the case of Aa1×nmatrix, is equivalent to Abeing a multiple of the matrix (1 ...1). In this paper, we treat the case when Ais a matrix defined by one row, and it does not verify this condition. The two main results are theorems 3.3 and 4.2; we find that, for matrices with at least three different elements the system always has a slope defined with respect to a hyperplane. We also calculate all the slopes with respect to the coordinate hyperplanes at the origin. The case with only two different elements was studied in [6]: in this case if the semigroup defined by the columns of Ais reduced, we have always a slope with respect to a hyperplane; when this semigroup is not reduced, we must do a change of coordinates and find a slope at infinity. This work generalizes the paper [3] by Castro and Takayama which treats the case of hypergeometric systems defined by one row, with a matrix of the following form (1 a2... a n), with 1 <a 2<··· <a n. In [3] they used a theorem of Laurent and Mebkhout [11] about restrictions and slopes. They need to compute the restriction of the hypergeometric system. In this process it is very useful that a1=1. Ifnot, the computation of the b-function (necessary in the known algorithms to compute restrictions) is very hard. On the other hand, to use the theorem it is also necessary that the calculation of the slopes in the new D-module obtained must be easier to do, but if a1= 1, it is possible that the presentation of the new module is in Dr, with r>1. To generalize this paper we need to use different techniques. 2. Filtrations and slopes The rings of differential operators Aand Dadmit several filtrations (the analogous to graduations in non-commutative rings). In this section we define the F-filtration or filtration by the order of the operator, the V-filtration or Malgrange- Kashiwara filtration, and a family of filtrations defined as a combination of these. We give the definition of the filtrations with respect to the ring D. Analogous definitions are valid in the Weyl algebra. With these objects we can formally define what a slope of a D-module is. GivenanoperatorP∈D,P=0, P= β aβ(x)∂β. We define the F-order of this operator as ordF(P)=max{|β|,a β=0} and ordF(0) = −∞. 4636 MAR´ IA ISABEL HARTILLO-HERMOSO For any kwe define the vector subspaces of the operators of order kor less than k, and the operators of order less than k, and we denote them as Fk(D)={P∈D,such that ordF(P)≤k}, F<k(D)={P∈D,such that ordF(P)<k}. This allows us to define the graded algebra grF(D)= k grF k(D)= k Fk(D)/F<k(D). For any operator Pwe can define its F-symbol, an element in the graded ring, as σF(P)=  β, |β|=ordF(P) aβ(x)ξβ. GivenanidealIin Dwe define the graded ideal grF(I)=σF(P),P∈I. In this case grF(D)∼ =C{x}[ξ]andgr F(A)∼ =C[x, ξ]. In an analogous way we can define the V-filtration or Malgrange-Kashiwara filtration, which is defined with respect to a hypersurface. If we consider the hypersurface xj= 0, for a non-zero operator P, P= α,β aα,β xα∂β. We define its V-order as ordV(P)=max{βj−αj,a α,β =0}, and ordV(0) = −∞. The vector subspaces are defined as Vk(D)={P∈D,such that ordV(P)≤k}, V<k(D)={P∈D,such that ordV(P)<k}. Then, the V-graded ring is grV(D)= k grV k(D)= k Vk(D)/V<k(D). For an operator we can define its V-symbol as σV(P)=  (α,β),β j−αj=ordV(P) aα,βxαξβ. As before we can define the graded ideal. The great difference in this case is that the graded ring is not commutative as before. Indeed grV(D)∼ =C[xj]{x1,...,x j−1,x j+1,...,x n}∂ with the usual relations [xi,x j]=0,[∂i,∂ j] = 0, and [∂i,x j]=δi,j.FortheWeyl algebra, grV(A)∼ =A. Using the above filtrations Fand V(with respect to xj= 0), we can define an ordered family of filtrations. Given (p, q)=(0,0), non-negative integers, we define the linear form over Q2,givenbyL(a, b)=pa +qb. Then, given an operator as before, we define the L-order of a non-zero operator: ordL(P)=max{L(|β|,β j−αj),a α,β =0}, IRREGULAR HYPERGEOMETRIC SYSTEMS 4637 as usual ordL(0) = −∞. We also define the vector subspaces: Lk(D)={P∈D,such that ordL(P)≤k}, L<k(D)={P∈D,such that ordL(P)<k}. The L-symbol of an operator is defined as σL(P)=  (α,β),L(|β|,βj−αj)=ordL(P) aα,βxαξβ. As before, we define the L-graded ideal and the graded ring grL(D). If p=0,then grL(D)∼ =C{x}[∂]andgr L(A)∼ =C[x, ∂]. We can always consider the F-filtration as an L-filtration for (p, q)=(1,0), and also the V-filtration as the L-filtration for (p, q)=(0,1). In general, to simplify notation, we shall write L=pF +qV to note the filtration Ldefined as before. We can define an order among the L-filtrations. Given two L-filtrations, L1 defined by (p, q)andL2by (p,q), L1<L 2if −p/q < −p/q.GivenaL-filtration we define its slope as the ratio −p/q. Definition 2.1. Given a left ideal Iand an L-filtration L=V, we define the L-characteristic variety, denoted ChL(D/I), as the analytic variety of C2ndefined by its graded ideal grL(I), i.e., ChL(D/I)={(x, ξ)∈C2n|σL(P)(x, ξ)=0,for all P∈I}. If the left ideal Iis in the Weyl algebra we define similarly the L-characteristic variety. Definition 2.2 ([9]).Let Ibe an ideal of D. The slopes of the D-module D/I with respect to xj= 0 are the slopes of the linear forms L=F, V (with Vdefined with respect to xj= 0) such that grL(I) is not bihomogeneous for the Fand V filtrations. In the Weyl algebra, we can define a filtration for a given vector (u, v)∈R2n such that ui+vi≥0: ord(u,v)(P)=max{αu +βv, aα,β =0}, Wk={P∈Asuch that αu +βv ≤k},W <k ={P∈Asuch that αu +βv < k}, gr(u,v)(A)= k grW k(A)= k Wk(A)/W<k(A), in(u,v)(P)=  (α,β)αu+βv=ord(u,v)(P) aα,βxαξβ. For this weight vector we shall use the notation of initial ideals: in(u,v)(I)=in(u,v)(P),P∈I. It is clear that, if we consider the filtration L=pF +qV , defined with respect to the hyperplane xi=0,then grL(I)=in (0,...,0,−q,0,...,0,p,...,p,p+q,p,...,p)(I) with −qin the i-th coordinate and p+qin the i+n-th coordinate. 4638 MAR´ IA ISABEL HARTILLO-HERMOSO 3. Hypergeometric systems defined by a reduced semigroup In this section, we consider a 1 ×ninteger matrix A=(a1a2... a n), such that the semigroup defined by its elements is reduced or, equivalently, such that all the elements aiare positive. First of all we introduce some notation and a theorem from [14]. Here we do not consider that the ideal IAis homogeneous: the result is true without this hypothesis. Definition 3.1 ([14]).Given a d×ninteger matrix Aand a complex vector β∈Cd, we define the (u, v)-fake initial ideal, when ui+vi≥0, as fin(u,v)(HA(β)) = inv(IA)+in(u,v)(Aθt−β), where in(u,v)(f) denotes the initial terms of f, or the initial ideal, with respect to the weight vector (u, v). In general we have the following result: Theorem 3.2 (Thm. 4.3.5 of [14]).Given a d×ninteger matrix Aand a complex vector β∈Cd, for a fixed weight vector (u, v), such that ui+vi>0,ifin(u,v)(Aθt−β) forms a regular sequence over the ring gr(u,v)(A)/inv(IA),then in(u,v)(HA(β)) = fin(u,v)(HA(β)). Proof. We consider as in [14] the following exact sequence of modules over the algebra gr(u,v)(A)/inv(IA): d  i=1 gr(u,v)(A)/inv(IA)·ei ¯ d1 →gr(u,v)(A)/inv(IA)→gr(u,v)(A)/fin(u,v)(HA(β))→0, where ¯ d1(n i=1 Piei)=n i=1 Piin(u,v)((Aθt−β)i). Since in(u,v)((Aθt−β)j)ei−in(u,v)((Aθt−β)i)ejclearly belongs to the kernel of ¯ d1, we can extend the exact sequence to the Koszul complex Kβ •(gr(u,v)(A/IA)): ··· ¯ d2 −→ Kβ 1(gr(u,v)(A/IA)) ¯ d1 −→ Kβ 0(gr(u,v)(A/IA)) −→ 0, where Kβ p(gr(u,v)(A/IA)) =  1≤i1<···<ip≤n gr(u,v)(A/IA)ei1···ip and ¯ dp(ei1···ip)= p  r=1 (−1)r−1in(u,v)((Aθt−β)ir)ei1···  ir···ip. We can also define the Koszul complex (with non-homogeneous elements) Kβ •(A/IA)as ···−→Kβ 2(A/IA)d2 −→ Kβ 1(A/IA)d1 −→ Kβ 0(A/IA)−→ 0, where Kβ p(A/IA)=  1≤i1<···<ip≤n A/IAei1···ip and dp(ei1···ip)= p  r=1 (−1)r−1((Aθt−β)ir)ei1···  ir···ip. IRREGULAR HYPERGEOMETRIC SYSTEMS 4639 This complex can be filtered by Fq(Kβ p(A/IA)) =  1≤i1<···<ip≤n Fq−p k=1 cikA/IAei1···ip, where cik=ord (u,v)((Aθt−β)ik). Clearly Kβ •(gr(u,v)(A/IA)) = gr(u,v)(Kβ •(A/IA)). Using this filtration it is easy to see that the sequence Kβ 1(gr(u,v)(A/IA)) −→ Kβ 0(gr(u,v)(A/IA)) −→ gr(u,v)(HA(β)) −→ 0 is exact if H1(Kβ •(gr(u,v)(A/IA))) = 0. If this sequence is exact, we have proved the statement, so all we need to prove is that H1(Kβ •(gr(u,v)(A/IA))) = 0, or it is enough to see that the elements in {in(u,v)((Aθt−β)1),...,in(u,v)((Aθt−β)d)}form a regular sequence in the commutative graded ring gr(u,v)(A)/inv(IA).  First, we consider a1<a 2<···<a n, and we study a generalization of the case considered in [3], where the condition a1= 1 was added. Theorem 3.3. Given A,a1×nmatrix, and β∈C, such that A=(a1a2... a n), with 0<a 1<a 2<··· <a n, the hypergeometric system HA(β)has no slopes with respect to the hyperplanes xi=0,i<n, and has only the slope an−1 an−1−anwith respect to xn=0. The singular locus of the system is {xn=0}. (i) If we consider a filtration L=pF +qV ,p>0,withVdefined with respect to xi=0,i<n,then ChL(HA(β)) = V(ξ1,...,ξ n−1,x nξn). (ii) If we consider a filtration L=pF +qV ,p>0,withVdefined with respect to xn=0,withL<a n−1F+(an−an−1)V,then ChL(HA(β)) = V(ξ1,...,ξ n−1,x nξn). (iii) If L>a n−1F+(an−an−1)V,then ChL(HA(β)) = V(ξ1,...,ξ n−2,ξ n,x n−1ξn−1). Proof. In first place we shall prove the first part of the theorem. Given A= (a1a2... a n), we consider the set of elements in the toric ideal IA: Ci,j =∂aj i−∂ai j,i<j. First of all, we prove that if we consider the hyperplane xj=0withj<n,there are not slopes of the system with respect to this hypersurface. Let Lbe a filtration defined with respect to xj= 0, such that L=pF +qV ,withp= 0. We consider the operators Ck,n.Ifk=j,thenσF(Cj,n)=ξan j, so it is clear that σL(Cj,n)=ξan j. If k=j,wehavethatσL(Ck,n)=σF(Ck,n)=ξan k. Finally, {ξan 1,ξan 2,...,ξan n−1}⊂grL(HA(β)) ⇒{ξ1,ξ 2,...,ξ n−1}⊂grL(HA(β)). Now, if we consider the element Q=a1x1∂1+a2x2∂2+···+anxn∂n−β,wehave σL(Q)=a1x1ξ1+a2x2ξ2+···+anxnξn∈grL(HA(β)), 4640 MAR´ IA ISABEL HARTILLO-HERMOSO so {ξ1,ξ 2,...,ξ n−1,x nξn}⊂grL(HA(β)), and Ldoes not define a slope of the system. This fact comes from the property of the module HA(β) being holonomic; see [1]. So for L=Vwe have that grL(HA(β)) is holonomic too, and then the L-characteristic variety can be decomposed in irreducible components, all of dimension n, and each of those components is included in some of those defined before by {ξ1,ξ 2,...,ξ n−1,x nξn}. But all these components are bihomogenous, hence Ldoes not define a slope. For further details see [5]. We prove now, at the same time, parts (i) and (ii). We fix the hyperplane xn= 0. We choose a filtration Ldefined as L=pF +qV , such that L<a n−1F+ (an−an−1)V(thesameresultsaretrueinthe(i)case).WehavethatσL(Ci,n)= ξan i, and the operators {C1,n,C 2,n,...,C n−1,n}are a Q-basis of ker(A). We are interested in the toric ideal, IA. Due to the fact that all ai>0, we have that the toric ideal IAis homogeneous with respect to the grading defined by A.If we note  IAthe ideal generated by Ci,n, then it is also graded, and  IA⊂IA. To compute a set of generators of IAwe use the ideal quotients. Using lemma 12.1 of [15] we can obtain ( IA:∂∞ i). First of all, we calculate ( IA:∂∞ n)sothat, choosing the monomial order ≺, graded with respect to this grading such that ∂1∂2···∂n,theset{C1,n,...,C n−1,n}forms a Gr¨obner basis (with respect to such a monomial ordering) of  IA. In the same way, if we fix an i<n,wecan consider the new monomial ordering ≺, defined as the graded monomial ordering such that ∂1∂2··· ∂i−1∂i+1 ··· ∂n∂i. So, finally we obtain (C1,n,C 2,n,...,C n−1,n:(∂1···∂n)∞)=C1,n,C 2,n,...,C n−1,n. Both ideals IAand  IAare saturated, and  IA⊂IA.Wehavedefined IAfrom a Q-basis of ker(A), hence we have that for any ∂u+−∂u − ∈IAthere always exists an element ∂v+−∂v − ∈ IA,andana∈Nsuch that au=v. On the other hand, it is clear that {ξan 1,ξan 2,...,ξan n−1} forms a regular sequence in C[ξ], so using proposition 4.3.2 of [14] we have that ξan 1,ξan 2,...,ξan n−1=in (p,...,p,p+q)( IA). This implies, with the above properties, ξan 1,ξan 2,...,ξan n−1=in (p,...,p,p+q)( IA)⊂in(p,...,p,p+q)(IA) ⊂ξ1,ξ 2,...,ξ n−1=in(p,...,p,p+q)( IA)=in(p,...,p,p+q)(IA). We want to prove that σL(a1x1∂1+···+anxn∂n−β) forms a regular sequence over C[x, ξ]/grL(IA), that is, this element is not a zero divisor in this ring. But, if we prove that this element is not a zero divisor over C[x, ξ]/grL( IA), we obtain the desired result. The ring C[x, ξ]/grL( IA) is a graded ring; we must see that σL(Aθ −β)isnota zero divisor, that is, if hσL(Aθ −β)∈grL( IA), then h∈grL( IA). But grL( IA)isa monomial graded ideal, so all the monomials in hxnξn∈grL( IA), thus h∈grL( IA). IRREGULAR HYPERGEOMETRIC SYSTEMS 4641 So, we have proved that, for any Lfiltration, with L<a n−1F+(an−an−1)V, grL(HA(β)) = fin(0,...,0,−q,p,...,p,p+q)(HA(β)) =in (p,...,p,p+q)(IA)+σL(a1x1∂1+···+anxn∂n−β) and it is clear that this is also true for any filtration Ldefined with respect to the Malgrange-Kashiwara filtration with respect to a hypersurface xi=0withi<n. In particular we have that ChL(HA(β)) = Vfin(0,...,0,−q,p,...,p,p+q)(HA(β) =Vin(p,...,p,p+q)(IA)+σL(a1x1∂1+···+anxn∂n−β) =V(ξ1,...,ξ n−1,x nξn). Finally, we prove part (iii). We now consider a new filtration Ldefined with respect the Malgrange-Kashiwara filtration, with respect to the hypersurface xn= 0, such that L>a n−1F+(an−an−1)V. In this case, we obtain that in(p,...,p,p+q)( IA)=ξan 1,ξan 2,...,ξan n−2,ξan−1 n. Using the same argument as before, we have that σL(a1x1∂1+···+anxn∂n−β) is also a regular sequence over C[x, ξ]/in(p,...,p,p+q)(IA), so grL(HA(β)) = fin(0,...,0,−q,p,...,p,p+q)(HA(β)) =in (p,...,p,p+q)(IA)+σL(a1x1∂1+···+anxn∂n−β). We want to calculate the L-characteristic variety as before. In this case, ChL(HA(β)) = V(ξ1,ξ 2,...,ξ n−2,ξ n,x n−1ξn−1). This change in the characteristic variety tells us that L=an−1F+(an−an−1)V is a slope of the hypergeometric system. It is also clear, as before, that any other filtration Lis not a slope of our system.  In theorem 3.3, we had the hypothesis a1<a 2<···<a n.Wenowdescribethe general case. Given a 1 ×ninteger matrix, which defines a reduced semigroup, we can always order its elements. We consider, in the first place, that one of the elements in the matrix is repeated, and the repeated element is not the largest one. To simplify notation, we suppose that a1is the repeated element, and it is written twice. It is clear that it can be easily generalized. So, A=(a1a1a2a3··· an−1)a1<a 2<···<a n−1. If we consider the following elements in the toric ideal: C1,n =∂an−1 1−∂a1 n,C i,n =∂an−1 i−∂ai−1 n,2≤i≤n−1, they form a Qbasis of ker(A). Repeating the process in the proof of theorem 3.3, we obtain that the L-symbol of the homogeneity operator forms a regular sequence, for every filtration L=an−2F+(an−1−an−2)V. We see that theorem 3.3 can be rewritteninthiscase. On the other hand, if the repeated element is the largest one, and we suppose that it is written twice, we have A=(a1a2... a n−1an−1),a 1<a 2<···<a n−1.