Singularities of the hypergeometric system associated with a monomial curve

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 9, Pages 3761–3775 S 0002-9947(03)03300-2 Article electronically published on May 29, 2003 SINGULARITIES OF THE HYPERGEOMETRIC SYSTEM ASSOCIATED WITH A MONOMIAL CURVE FRANCISCO JES´ US CASTRO-JIM´ ENEZ AND NOBUKI TAKAYAMA Abstract. We compute, using D-module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems. 1. Introduction Let An=Chx1,...,x n,∂ 1,...,∂ nibe the Weyl algebra of order nover the complex numbers Cand let C[∂]=C[∂1,...,∂ n] be the subring of Anof linear differential operators with constant coefficients. Let A=(aij ) be an integer (d×n)-matrix of rank d.WedenotebyIA⊂C[∂] the toric ideal associated to A: i.e., IAis the ideal generated by the set {∂u−∂v|u, v ∈Nn,Au T=AvT} where ()Tmeans “transpose”. We denote by θthe vector (θ1,...,θ n)Twith θi=xi∂i. Foragivenβ= (β1,...,β d)T∈Cdwe consider the column vector (in Ad n)Aθ −βand we denote by hAθ −βithe left ideal of Angenerated by the entries of Aθ −β. Following Gel’fand, Zelevinski˘ı and Kapranov [6], we denote by HA(β)theleft ideal of Angenerated by IA∪hAθ−βi. It is called the GKZ-hypergeometric system associated to the pair (A, β). The quotient HA(β)=An/HA(β) is a holonomic Anmodule (see e.g. [17]). If the toric ideal IAis homogeneous, i.e., if the Q-row span of Acontains (1,...,1), it is known ([8]; see also [17]) that HA(β) is regular holonomic and the book [17] is devoted to an algorithmic study of such systems. Especially, the book gives an algorithmic method to construct series solutions around singular points of the system. In this article, we start a study of singularities of GKZ-hypergeometric systems for non-homogeneous toric ideals IAby treating the “first” case when d=1,A= (a1,a 2,...,a n)∈Zn,a1= 1. We evaluate the geometric slopes of HA(β)by successive restrictions of the number of variables. The slopes characterize Gevrey class solutions around a singular locus. Let us remark on an analytic meaning of slopes. Let X=Cnand Y={x|xn=0}⊂X.WedenotebyOd X|Ythe Received by the editors November 15, 2002. 2000 Mathematics Subject Classification. Primary 32C38, 13N10; Secondary 13P10, 14F10, 14M25. Key words and phrases. Algebraic geometry, D-modules, toric varieties, hypergeometric systems. The first author was partially supported by BFM-2001-3164, FQM-218 and FQM-813. c 2003 American Mathematical Society 3761 3762 FRANCISCO JES´ US CASTRO-JIM´ ENEZ AND NOBUKI TAKAYAMA formal completion of OXalong Y(i.e., series that are formal in xnand convergent in x1,...,x n−1). To each real number s∈[1,+∞)wedenotebyOX|Y(s)the subsheaf of Od X|Yof Gevrey functions of order s(along Y). The sheaf OX|Y(1) is the restriction OX|Yand, by definition, we write OX|Y(+∞)=Od X|Y. For any holonomic DX-module M, Z. Mebkhout [14] associates the sheaf IrrY(s)(M)as the solution sheaf RHomD(M,OX|Y(s)/OX|Y). One fundamental result in the irregularity of D-modules is the fact that IrrY(s)(M) is a perverse sheaf, for any s (see [14]). These sheaves define a filtration of the irregularity of Malong Y, i.e., IrrY(M):=Irr Y(+∞)(M). The main result of [11] is that 1/(1 −s)isaslope of Mwith respect to Yif and only if sis a gap of the graduation defined by the filtration on the irregularity. In other words, 1/(1 −s) is a slope if and only if IrrY(s)(M)/IrrY(<s)(M)6=0. Our evaluation of the slopes is done as follows: (1) We translate Laurent and Mebkhout’s theorem [12] on restrictions and slopes of D-modules into an algorithm to evaluate the slopes by utilizing the results of [2] and [15]. (2) Apply our general algorithm to the hypergeometric system associated to A=(1,a 2,...,a n). This system has many nice properties and our algorithm outputs the slopes without computation on computers. In the last section we study rational solutions and reducibility of our systems. 2. Micro-characteristic varieties In this section, following Laurent [10], we describe micro-characteristic varieties for a given D-module. We will state a result of Laurent and Mebkhout [12, Corollaire 2.2.9] (see also [14, p. 125] and [11, p. 42]), allowing to reduce our general problem of evaluating the slopes to fewer variables. In this section X=Cnand DX=Dis the sheaf of linear differential operators with holomorphic function coefficients. Let Mbe a coherent D-module. Recall that the characteristic variety of M(denoted by Ch(M)) is an analytic subvariety of the cotangent bundle T∗X. Suppose that Y⊂Xis a smooth hypersurface. We say that Yis non-characteris- tic for Mif T∗ YX∩Ch(M)⊂T∗ XX.HereT∗ YXis the conormal bundle to Yin X and T∗ XXis the zero section of T∗X. Now, following Laurent [9], [10], we shall define the notion of non-micro-charac- teristic variety for M. To simplify the presentation we will assume that (x1,x 2,..., xn) are local coordinates in Xand that Yis defined by xn=0. Wedenoteby (x1,...,x n,ξ 1,...,ξ n) local coordinates in T∗X. Sometimes it will be useful to write x1=y1,...,x n−1=yn−1,xn=t,ξ1=η1,...,ξ n−1=ηn−1and ξn=τ. Let us denote by Λ the conormal bundle of Yin T∗X(i.e., Λ = T∗ YX). So, in local coordinates, Λ = {(y,t,η,τ)∈T∗X|t=η=0}. Here, η=(η1,...,η n−1)and y=(y1,...,y n−1). We denote by (y,τ,y∗,τ∗) local coordinates on the cotangent bundle T∗Λ. We denote by V•(D)(orsimplybyV) the Malgrange-Kashiwara filtration associated to Yon Dand by F•(D)(orsimplybyF) the order filtration on D.Fora given rational number p/q ≥0wedenotebyLp/q the filtration on Ddefined by pF +qV .TheLp/q-order of a monomial yαtl∂β y∂k tis equal to p(|β|+k)+q(k−l), where |β|=β1+···+βn−1. We will simply write L=Lp/q if no confusion arises. For p>0 the associated graded ring grL(D) is canonically isomorphic to π∗O[T∗Λ] [10, p. 407], where π:T∗Λ→Λ is the canonical projection and O[T∗Λ] denotes SINGULARITIES OF HYPERGEOMETRIC SYSTEMS 3763 holomorphic functions on T∗Λ that are polynomials on the fibers of π.Inlocal coordinates, grL(D0) is expressed as C{y}[τ,y∗,τ∗]whereD0is the stalk of Dat the origin. Given a differential operator P=X αlβk pαlβkyαtl∂β y∂k t, the L-order of Pis the maximun value of p(|β|+k)+q(k−l) over the monomials of P.Forp>0 we define the L-principal symbol of Pby σL(P)=Xpαlβkyα(τ∗)l(y∗)β(−τ)k where the sum is taken over monomials with maximal L-order. The L-principal symbol of Pis an element of grL(D) and then is a function on T∗Λ. In the classical case, i.e., for L=F,gr F(D)isidentifiedwithC{x}[ξ1,...,ξ n]=C{y,t}[η,τ]and the F-principal symbol of Pis simply denoted by σF(P)=Ppαlβkyαtlηβτkwhere the sum is taken for |β|+kmaximum. To each left ideal I⊂Dwe denote by σL(I)theidealofgr L(D) generated by the set of σL(P)forP∈I. For each L-filtration on Dwe associate a “good” L-filtration on M,bymeans of a finite presentation. The associated grL(D)-module grL(M) is coherent (see [10, 3.2.2]). The radical of the annihilating ideal AnngrL(DX)(grL(M)), which is independent of the “good” filtration on M, defines an analytic subvariety of T∗Λ. This variety is called the L-characteristic variety of Mand it is denoted by ChL(M). Suppose now that Z⊂Xis a smooth hypersurface transverse to Y. Suppose for simplicity that Zis defined in local coordinates by y1= 0. The conormal space Λ0:= T∗ Y∩ZZis a smooth subvariety of Λ = T∗ YXdefined in local coordinates by y1=0. SoT∗ Λ0Λ is the subvariety of T∗Λ defined in local coordinates by y1=y0∗ =τ∗=0,wherey0=(y2,...,y n−1). Definition 2.1 ([9], [12]).We say that Zis non-micro-characteristic of type L for Mif T∗ Λ0Λ∩ChL(M) is contained in T∗ ΛΛ. Sometimes we will say that, if this condition holds, Zis non-L-micro-characteristic for M. The sheaf of rings grL(D) is endowed with two graduations: The first is induced by the F-filtration and the second is induced by the V-filtration. Recall Laurent’s definition of slope of a coherent D-module M. Definition 2.2 ([10]).The rational number −p/q is said to be a slope of Mwith respect to Yat the origin if and only if the radical of the ideal AnngrL(DX)(grL(M)) is not bihomogeneous for F-norV-graduations. An important consequence of the work [11] (see Th´eor`eme 2.4.2) is what follows: A holonomic DX-module Mis regular with respect to Yat the origin if and only if Mhas no slope with respect to Yat the origin. We will use this fact freely in the text. Remark. In [12], ∞and 0 (Fand V) were included in the set of the slopes. We do not include them in the set of the slopes in this paper. 3764 FRANCISCO JES´ US CASTRO-JIM´ ENEZ AND NOBUKI TAKAYAMA Finally, the following result by Laurent and Mebkhout allows induction on the number of variables to calculate slopes. Theorem 2.3 ([12, Corollaire 2.2.9]).Let Mbe a holonomic DX-module. Let Zand Ybe transverse smooth hypersurfaces on Xsuch that Zis non-L-micro- characteristic (for all p>0,q>0) for M. Then the slopes of Mwith respect to Y equal the slopes of M0with respect to Y0=Z∩Y,whereM0is the restriction of Mto Z. This is a deep result in D-module theory. Its proof uses the algebraic-analytic comparison theorem ([11, Theorem 2.4.2]) and a Cauchy-Kowalewska theorem for Gevrey functions with respect to Z([12, Corollaire 2.2.4]; see also [14, Th´eor`eme 6.3.4]). 3. Computing slopes by reducing the number of variables We have introduced the notion of the slopes and the invariance of them under restrictions satisfying a condition on L-characteristic varieties. In the sequel, we assume that our ideal is that of the Weyl algebra An. Constructions in sheaves such as restrictions and L-characteristic varieties in the previous section can be done via constructions in the Weyl algebra as we usually see in the computational D-module theory. We are interested in computation of the slopes. The slopes of An/I (at the origin) along xn= 0 can be computed by the ACG algorithm introduced in [2]. In this section, translating Laurent and Mebkhout’s result into computer algebra algorithms, we will give a preprocessing method for the ACG algorithm to accelerate the original. The preprocessing is useful for a class of inputs including GKZ hypergeometric ideals as we will see in Section 4. Let us first recall the ACG algorithm. A weight vector is an element W=(u1,...,u n,v 1,...,v n)∈R2nsuch that ui+vi≥0 for all i. This weight vector W=(u, v) induces a natural filtration on An, and it is called the W-filtration. The associated graded ring is denoted by grW(An), and for each left ideal I⊂Anthe associated graded ideal is denoted by inW(I)orin (u,v)(I). Here, the initial ideal in(u,v)(I) is the ideal generated by in(u,v)(f),f ∈Iin grW(An). When ui+vi>0, it is an ideal in the polynomial ring of 2nvariables: grW(An)= C[x1,...,x n,ξ 1,...,ξ n]. The initial ideal of Iwith respect to the weight (u, v) is generated by the (u, v)-initial terms of a Gr¨obner basis of Iby an order that refines the partial order defined by (u, v). See, e.g., [17, Theorem 1.1.6]. Consider the filtration L=pF +qV,p > 0,q > 0 introduced in the previous section. The ideal σL(I), which gives the L-characteristic variety, can be expressed in terms of the initial ideal as follows: σL(I)=gr L(D)·in`(I)|x17→y1,...,xn−17→yn−1,xn7→τ∗,ξ17→y∗ 1,...,ξn−17→y∗ n−1,ξn7→−τ, `=p( n z}| { 0,...,0, n z}| { 1,...,1) + q( n z}| { 0,...,0,−1, n z}| { 0,...,0,1). SINGULARITIES OF HYPERGEOMETRIC SYSTEMS 3765 For two weight vectors Wand W0and a term order <,wedenoteby<W,W 0the order xα∂β<W,W 0xa∂b ⇔W·(α, β)<W·(a, b) or W·(α, β)=W·(a, b)andW0·(α, β)<W0·(a, b) or W∗·(α, β)=W∗·(a, b)forbothW∗=W, W 0and xα∂β<x a∂b. To each differential operator P=Ppαβxα∂β∈Anwe associate the Newton polygon N(P)ofP(with respect to xn= 0) defined as the convex hull of the subset of Z2,[ pαβ 6=0 (|β|,β n−αn)+(−N)2. Let Ibe a left ideal in An. As we said in Definition 2.2, the notion of slope of a differential system was introduced by Y. Laurent [10]. Let us give here a slightly different but equivalent definition: the number r,−∞ <r<0, is a geometric slope of I(or of An/I) with respect to xn=0ifandonlyifpσ(−r)F+V(I)isnot bihomogeneous with respect to the weight vectors F=(0,...,0,1,...,1) and V= (0,...,0,−1,0,...,0,1). Following [2], we say that the number r,−∞ <r<0is an algebraic slope of I(or of An/I) if and only if σ(−r)F+V(I) is not bihomogeneous with respect to the weight vectors Fand V. The geometric slope is simply called the slope in this paper if confusion does not arise. Note that we may consider inL(I) instead of σL(I) so long as we are concerned about homogeneity. For algebraic or geometric slope r, the weight vector L=(−r)F+Vlies on a face of the Gr¨obner fan of I([3], [17]), which yields the following algorithm. Algorithm 3.1 ([2], ACG algorithm). Input: G={P1,...,P m}(generators of an ideal I). Output: All algebraic and geometric slopes of An/I with respect to xn=0atthe origin. geometric slope =∅;algebraic slope =∅; F=(0,...,0,1,...,1); V=(0,...,0,−1,0,...,0,1); p=1;q=0;slope =−∞;previous slope =slope; while (slope!=0){ L=pF +qV; G=aGr¨obner basis of Iwith respect to the order <L,V ; slope = the minimum of 0 and {the slopes rof the Newton polygon N(P)|P∈G, r>previous slope} if slope =0,then return (algebraic slope and geometric slope). if σL(G) is not homogeneous for Fnor Vthen { algebraic slope =algebraic slope ∪{slope} } if pσL(G) is not homogeneous for Fnor Vthen { geometric slope =geometric slope ∪{slope} } p= numerator(|slope|); q= denominator(|slope|); previous slope =slope; } return(algebraic slope and geometric slope) 3766 FRANCISCO JES´ US CASTRO-JIM´ ENEZ AND NOBUKI TAKAYAMA Figure 1. Here, we use the convention F:= ∞F+V. Note that the ideal J=hf1,...,f mi ⊂C[x1,...,x n,ξ 1,...,ξ n] is homogeneous for the weight (u, v)∈Z2nif and only if all (u, v)-homogeneous subsums of fibelong to the ideal J. For ordinary differential equations, the ACG algorithm is nothing but the wellknown Newton polygon method; see two examples below. Example 3.2. Put G={xp+1 1∂1+p},n=1,p∈N. Then, the ACG algorithm returns {−p}.(exp(x−p 1) is a classical solution of G.) Example 3.3. Put G={2x1(x1∂1)2+x2 1∂1+1},n= 1. Then, the ACG algorithm returns {−1/2}.(exp(x−1/2 1) is a classical solution of G.) The next example of two variables is generated by the computer algebra system kan/k0 [18]. Example 3.4. We consider the GKZ hypergeometric ideal Iassociated to the matrix A=(1,3) and β=−3. We will compute the slopes of A2/I at the origin along x2= 0 by the ACG algorithm. The Gr¨obner basis of Ifor the weight ( x1 z}|{ 0, x2 z}|{ 0, ∂1 z}|{ 1, ∂2 z}|{ 1)is [ x1*Dx1+3*x2*Dx2+3, -Dx1^3+Dx2, -3*x2*Dx1^2*Dx2-5*Dx1^2-x1*Dx2, -9*x2^2*Dx1*Dx2^2-36*x2*Dx1*Dx2+x1^2*Dx2-20*Dx1, -27*x2^3*Dx2^3-189*x2^2*Dx2^2-x1^3*Dx2-276*x2*Dx2-60 ] Here, Dxiand xistand for ∂iand xi, respectively. The Newton polygons N(P)’s are given in Figure 1. From the Newton polygons, −1/2 is the candidate of the first slope. Next, we compute the Gr¨obner basis for the weight (0,−2,1,3) = (0,0,1,1) + 2(0,−1,0,1). The Gr¨obner basis is [ x1*Dx1+3*x2*Dx2+3, Dx2-Dx1^3 ] The radical is generated by [ -x1*Dx1-3*x2*Dx2, -Dx1^3+Dx2, 3*x2*Dx1^2*Dx2+x1*Dx2, -9*x2^2*Dx1*Dx2^2+x1^2*Dx2, 27*x2^3*Dx2^3+x1^3*Dx2 ] SINGULARITIES OF HYPERGEOMETRIC SYSTEMS 3767 Here, x1 =y1,x2 =τ∗,Dx1 =y∗ 1,Dx2 =−τ. It is not bihomogeneous, and then −1/2 is a geometric and algebraic slope. By looking at the two Newton polygons of x1*Dx1+3*x2*Dx2+3,Dx2-Dx1^3,we see that there are no slopes larger than −1/2. Then, the ACG algorithm terminates here. The ACG algorithm requires a repetition of Gr¨obner basis computations in the Weyl algebra of 2nvariables to evaluate the slopes of An/I. However, if xi=0 is non-micro-characteristic of type L=(−r)F+Vfor all −∞ <r<0andthe restriction of An/I to xi= 0 is singly generated, we can preprocess the input so that the input ideal for the ACG algorithm lies in the Weyl algebra of 2(n−1) variables. The correctness of the following algorithm can be shown by Laurent and Mebkhout’s Theorem 2.3. Algorithm 3.5 (Computing slopes with a preprocessing). Step 1: Check if xi= 0 is non-micro-characteristic of An/I for all types Lby calling Algorithm 3.6. Step 2: Compute the restriction of An/I to xi= 0 and check if it is expressed as D0/I0where D0is the Weyl algebra of 2(n−1) variables. Step 3: If we failed either in Step 1 or in Step 2, then apply the ACG algorithm for I. If we succeeded both in Step 1 and in Step 2, then try to reduce more variables or apply the ACG algortihm for I0. We can compute restrictions of a given D-module by using Oaku’s algorithm [15]. This algorithm is implemented in computer algebraic systems Macaulay2 and Kan [7], [18], [13]. Therefore, the remaining algorithmic question for the preprocessing is to determine the range of type L=(−r)F+Vfor which xi=0(i≤n−1) is non-micro-characteristic. It follows from the definition of non-micro-characteristic that the question is nothing but to find the segment (−∞,r 1) such that V(σ(−r)F+V(I),y∗ 1,...,y∗ i−1,y i,y∗ i+1,...,y∗ n−1,τ∗)⊆V(y∗ 1,...,y∗ n−1,τ∗)=T∗ ΛΛ for r∈(−∞,r 1). Here, V(f1,...,f m) is the affine variety defined by the polynomials f1,...,f m. The inclusion condition can be algebraically rephrased as qin(−r)F+V(I),ξ 1,...,ξ i−1,x i,ξ i+1,...,ξ n3ξi. Since the Gr¨obner fan is a finite union of Gr¨obner cones [2], the range can be determined by a similar method with the ACG algorithm. Algorithm 3.6. range of nonMC(H,r0) Input: His a finite set in An,r0is a negative number or −∞. Output: r1such that x1= 0 is non-micro-characteristic of type (−r)F+V for An/An·{H},forr∈[r0,r 1). previous slope =slope =r0; F=(0,...,0,1,...,1); V=(0,...,0,−1,0,...,0,1); while (slope!=0){ p= numerator(|slope|); q= denominator(|slope|); L=pF +qV; G=aGr¨obner basis of Hwith respect to <L; 3768 FRANCISCO JES´ US CASTRO-JIM´ ENEZ AND NOBUKI TAKAYAMA if phσL(H),y 1,y∗ 2,...,y∗ n−1,τ∗i3y∗ 1and qhσpF+(q+ε)V(H),y 1,y∗ 2,...,y∗ n−1,τ∗i3y∗ 1 then { previous slope =slope; slope = the minimum of 0 and {the slopes rof the Newton polygon N(P)|P∈G, r>previous slope} }else { return(slope); } } return(0); Here, εis a sufficiently small positive rational number so that pF +(q+ε)Vlies in the interior of a Gr¨obner cone. In the case when r0=−∞, we use the convention numerator(|r0|)=1and denominator(|r0|) = 0, and F-non-micro-characteristic means that it is non-char- acteristic in the classical sense. Example 3.7. Suppose n≥2. Then range of nonMC({∂2 1−∂2},−∞) returns 0. If the function range of nonMC(I,−∞) returns 0, then x1= 0 is non-micro- characteristic of type pL +qV (p>0, q≥0) for An/I. Wenotethatitisnot always a clever strategy to call the function with the full set of generators. In fact, if x1= 0 is non-micro-characteristic of type Lfor An/J,thenitisnonmicro-characteristic of type Lfor An/I for any I⊇J. Therefore, for Step 1, it is sometimes more efficient to call the function range of nonMC for a subset of the generators of the input ideal as we will see in the case of GKZ hypergeometric ideals in the next section. 4. Computing slopes of H(1,a2,...,an)(β) Put A=(1,a 2,...,a n), a1=1<a 2<···<a n. We will evaluate the slopes of the GKZ hypergeometric ideal HA(β) associated to the 1 ×nmatrix Aand β∈C by using the general algorithm given in Section 3. To apply this algorithm, we need to find non-micro-characteristic varieties and compute the restrictions of HA(β)to these varieties. When f1,...,f mare polynomials in C[x1,...,x n,ξ 1,...,ξ n], we denote by V(f1,...,f m) the affine subvariety in C2ndefined by the fi. The following theorem can be shown by a standard method of Koszul complex ([1], [6]). Theorem 4.1. The characteristic variety of HA(β)is V(ξ1,...,ξ n−1,x nξn).In particular, the singular locus of HA(β)is xn=0. Note that there is no slope along xi=0,1≤i≤n−1, which can be shown easily. Recall that F=(0,...,0,1,...,1) and V=(0,...,0,−1,0,...,0,1). For a positive number pand a non-negative number q, we define the weight vector L= pF +qV . In Section 2, we explained the notion of non-micro-characteristic. When the variety is yi=xi= 0, this notion is rephrased as follows: for a given left SINGULARITIES OF HYPERGEOMETRIC SYSTEMS 3769 An-module An/I, the hyperplane yi=0(1≤i≤n−1) is called non-micro- characteristic of type Lwhen qhσL(I),y i,y∗ j,(j6=i),τ∗i3y∗ i. Proposition 4.2. For the hypergeometric An-module HA(β), the variety yi=0 (1 ≤i≤n−2) is non-micro-characteristic of type Lfor all L=pF +qV ,p>0. Proof. Consider ∂aj i−∂ai j∈HA(β). For all Land for i<j≤n−1, we have σL(∂aj i−∂ai j)=(y∗ i)aj, which implies that yi= 0 is non-micro-characteristic of type L. Now, let us apply the second step of the algorithm to evaluate the slopes, i.e., we will compute the restriction of HA(β)toyi=xi=0(1<i≤n−2). Let sbe an indeterminate. Consider the ideal HA[s]inAn[s] generated by Aθ −sand IA. Theorem 4.3. We have a left D0[s]-module isomorphism (4.1) An[s]/(An[s]HA[s]+xiAn[s]) ≃D0[s]/D0[s]HA0[s],i6=1. Here, D0=Chx1,...,x i−1,x i+1,...,x n,∂ 1,...,∂ i−1,∂ i+1,...,∂ niand A0=(1,a 2, ...,a i−1,a i+1,...,a n). Proof. Fix an order ≺such that ∂i∂n∂n−1···∂1. Then, ∂n−∂an 1,∂ n−1−∂an−1 1, ...,∂ 2−∂a2 1 is the reduced Gr¨obner basis of IAwith respect to . For each i(2 ≤i≤n), by applying the same method as in the proof of [17, Theorem 3.1.3], we can prove that in(−ei,ei)(HA(s)) is generated by Aθ −sand inei(∂j−∂aj 1)(j=2,...,n). Define the Vi-filtration Fk[s]ofAn[s]by Fk[s]=nXaαβγ xα∂βsγ|(β−α)·ei≤ko. We compute the restriction of An[s]/HA(s)toxi= 0 by Oaku’s algorithm (see, e.g., [17, Theorem 5.2.6, Algorithm 5.2.8]). Since in(−ei,ei)(xi∂i−xi∂ai 1)=xi∂i, the b-function of HA(s)alongxi=0isb(p)=p. Therefore, by [17, Theorem 5.2.6], we have the following isomorphism of left D0[s]-modules: An[s]/(HA(s)+xiAn[s]) ≃F0[s]/ F0[s](Aθ −s)+ n X j=2,j6=i F0[s](∂j−∂aj 1)+F−1[s](∂i−∂ai 1)+xiF1[s]  ≃D0[s]/ D0[s](Aθ −s)+ n X j=2,j6=i D0[s](∂j−∂aj 1) .  We can specialize sto any complex number β. In fact, we have the following theorem. Theorem 4.4. An/(AnHA(β)+xiAn)≃D0/D0HA0(β),i6=1.