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Polynomial and Rational Solutions of Holonomic 0 0 0 Systems 2 n Toshinori Oaku, Nobuki Takayama and Harrison Tsai a J 2 January 9, 1999 1 ] G 1 Introduction A . Polynomial and rational solutions for linear ordinary differential equations can h be obtained by algorithmic methods. For instance, the maple package DEtools t a provides efficient functions polysols and ratsols to find polynomial and ra- m tional solutions for a given linear ordinary differential equation with rational [ function coefficients. A natural analogue of the notion of linear ordinary differential equation in 1 v theseveralvariablecaseisthenotionofholonomicsystem. Aholonomicsystem 4 isa systemoflinear partialdifferentialequationswhosecharacteristicvarietyis 6 middle dimensional. 0 Chyzak [4] gave an algorithm to find the rational solutions of holonomic 1 systems by using elimination in the ring of differential operators with rational 0 0 function coefficients combined with Abramov’s algorithm for rational solutions 0 of ordinary differential equations with parameters. To the authors, solving / holonomic systems is analogous to solving systems of algebraic equations of h t zero-dimensionalideals. Underthisanalogy,themethodofChyzakcorresponds a to the elimination method for solving systems of algebraic equations. m The aim of this paper is to give two new algorithms, which are elimination : v free, to find polynomial and rational solutions for a given holonomic system i associated to a set of linear differential operators in the Weyl algebra X r D =k x ,...,x ,∂ ,...,∂ a 1 n 1 n h i where k is a subfield of C. Polynomialandrationalsolutionscanbe obtained,iftheyexist,byusingan exhaustivesearch. Forinstance,whenf =0isthesingularlocusofaholonomic system M = D/I, any rational solution has the form g/fr. If we have upper bounds for the degree of the polynomial g and for r, then we can construct all rational solutions by solving linear equations satisfied by the coefficients of g. Alternatively,ifweknowthedimensionofrationalsolutions,thenwecanobtain all rational solutions by increasing the degree of g and r. Hence, the problem reduces to finding effective bounds for these numbers. 1 In sections 2 and 3, we give algorithms for upper bounds on the degree of g and on r. The main techniques we use are Gröbner deformations in D as introduced in the book [11] and the b-function for D/I and f. In section 4, we give an algorithm to evaluate the dimension of polynomial and rational solutions. Our approach is an analog in D of a question studied by Singer [12], who gave an algorithm to compute Hom (M,N) for left R R := k(x ) ∂ -modules M and N and studied its relation to factorizations of 1 1 h i ordinarydifferentialoperators. ThetheoryofD-modulestranslatesourproblem on polynomial and rational solutions to constructions in the ring of differential operators D. For example, the k-vector space Hom (D/I,k[x]) H−n(Ω L D(D/I)) D ≃ ⊗D isthespaceofthepolynomialsolutionsoftheleftidealI. Here,Ωisthemodule of the top dimensional differential forms and D is the dualizing functor. See, e.g., the book of Björk [3] on this translation. We evaluate the dimension of the right hand side by recent developments of computational algebra such as constructionof free resolutions in the ring D and restrictions of D-modules [8], [9], [14], [17]. Our method also allows us to evaluate the dimension of solutions to a holonomic system inside any holonomic module. For instance, we can find the dimension of the delta function solutions to I. Throughout the paper, we refer to the book [11] for fundamental facts on thealgorithmictreatmentofD. Also,thealgorithmswhichappearinthepaper have been implemented in either kan [13] or Macaulay 2 [5]. We deeply thank Dan Grayson, Anton Leykin, and Mike Stillman, who helpedtoimplementD-modulesinMacaulay2,andFrédéricChyzakandMichael Singer for discussions on rational solutions. 2 Polynomial solutions by Gröbner deformations How can we obtain all polynomial solutions for ordinary differential equations? One method is to compute the indicial polynomial at infinity, find an upper bound on the degrees of polynomial solutions, and determine the coefficients of polynomials. The analogous method works for holonomic systems by using Gröbner deformations. For ℓ D and the weight vector w Rn, we denote by ∈ ∈ in (ℓ) the initial term of ℓ with respect to the weight ( w,w) (see, e.g., (−w,w) − [11, 1.1]). The following proposition follows from the definition of in (ℓ). (−w,w) § Proposition 2.1 Suppose that f(x ,...,x ) is a polynomial solution of I = 1 n D ℓ1,...,ℓm . Take w Zn. Then f(tw1x1,...,twnxn) can be expanded as a ·{ } ∈ polynomial in t as f (x)tp+O(tp+1). w Then we have in (ℓ ) f =0. (−w,w) i w • 2 The initialidealin (I) is sometimes calledthe Gröbnerdeformationof (−w,w) I with respect to ( w,w). − Theorem 2.2 [2] There exist only finitely many Gro¨bner deformations. The Newton polytope of a polynomial solution f is defined as the convex hull of the exponent vectors of f. For generic w, f is a monomial cxa and the w point a is a vertex of the Newton polytope of f. Let R = k(x ,...,x ) ∂ ,...,∂ and θ = x ∂ . Since a Zn belongs to 1 n 1 n i i i h i ∈ the zero set of the indicial ideal in (I)=R in (I) k[θ ,...,θ ], (−w,w) (−w,w) 1 n · ∩ g we canconstructa polytope thatcontainsthe Newtonpolytopes ofthe polyno- mialsolutionsbytakingthe convexhullofallthe non-negativeintegralrootsof all possible indicial ideals. It is not necessary to find all Gröbner deformations to obtain polynomial solutions. Let b(s) be the generator of in (I) k[s], s = n w θ . The (−w,w) ∩ i=1 i i polynomial b(s) is called the b-function of I with respect to ( wP,w). The next − proposition follows from the definition of b(s). Proposition 2.3 Let w be a strictly negative weight vector. In other words, we assume that w < 0 for all i. Consider the b-function b(s) of I with respect to i ( w,w) and let k be the smallest integer root of b(s) = 0. The polynomial 1 − − solutions of I have the form c xp. (1) p pi≥0X,p·w≤k1 Algorithm 2.4 (Finding the polynomial solutions by a Gröbner deformation) Input: a holonomic left ideal I. Output: the polynomial solutions of I. 1. Take a strictly negative weight vector w, compute the Gröbner deformation in (I), and compute the smallest non-positive integer root k (−w,w) 1 − of the b-function with respect to ( w,w). See, e.g., [11, Alg. 5.15] for − these procedures. 2. If we do not have such a root, then there is no polynomial solution other than 0. 3. If there is aminimal integerroot,thendetermine the coefficientsc of(1) p by solving linear equations for the coefficients. 3 Example 2.5 The followingsystemofdifferentialequationsoftwovariablesis called the Appell differential equation F (a,b,b′,c) [1]: 1 θ (θ +θ +c 1) x(θ +θ +a)(θ +b), x x y x y x − − θ (θ +θ +c 1) y(θ +θ +a)(θ +b′), y x y x y y − − (x y)∂ ∂ b′∂ +b∂ x y x y − − where a,b,b′,c are complex parameters. Let us demonstrate how Algorithm 2.4 works for the system of parameter values (a,b,b′,c) = (2, 3, 2,5). First, − − we choose a strictly negative weight vector w = ( 1, 2) and compute the − − b-function b(s), s = θ 2θ , which is the generator of the principal ideal x y − − in (I) Q[ θ 2θ ]. We can use the V-homogenization or the homoge- (−w,w) x y ∩ − − nized Weyl algebra to get the generator (see, e.g., [11, 1.2]). Second, we need § to find the integer roots of the b-function b(s)=0. In our example, these are 7,0,4. − FromProposition2.1,thehighest( w)-degreemonomialcxpyq inapolynomial − solutiongivesrise to an integersolutionw p+w q = p 2q ofthe b-function. 1 2 − − Hence, the polynomial solutions are of the form f = c xpyq. pq p,q≥0X,p+2q≤7 Finally, we determine the coefficients c by applying the differential operators pq to f andputting the results to 0. In our example,we have only one polynomial solution 1 1 4 3 24 3 ( y2+ y )x3+( y2 y+ )x2 −21 7 − 35 14 − 35 5 12 6 6 1 4 + ( y2+ y )x+ y2 y+1. −35 5 − 5 5 − 5 3 Rational solutions by Gröbner deformations The singular locus of a D-ideal I is defined to be the projection of the char- acteristic variety of I minus the zero section from the cotangent bundle to the coordinate base space. In other words, it is the zero set Sing(I)=V in (I):(ξ ,...,ξ )∞ k[x ,...,x ] . (0,e) 1 n 1 n h i∩ (cid:0) (cid:1) Any rational solution to I has its poles contained inside the singular locus. Thus if f(x) defines the codimension1 componentofSing(I), we may limit our search for rational solutions to k[x][1]. f Wewillpresentamethodtoobtainanupperboundoftheorderofthepoles alongf =0foreachrationalsolution. Forthispurposeweusethenotionofthe 4 b-functionforf andasectionuofaholonomicsystem,whichwasintroducedby Kashiwara[7]: Let bethesheafofalgebraicdifferentialoperatorsonX =Cn. D Foraholonomic -module = / I andapolynomialf,considerthetensor D M D D product = [f−1,s]fs . (2) N O ⊗OX M This has a structure of a left -module via the Leibnitz rule. Let u be a N D section of . Then the b-function for f and u (or for fsu) at p Cn is the M ∈ minimum degree monic polynomial 0=b(s) C[s] such that 6 ∈ b(s)fs u [s](fs+1 u) (3) ⊗ ∈D ⊗ holdsin atp(i.e.,asagermof atp). Thisb-functiondependsonthepoint N N p. Asafunctionofp,thereisastratificationofCn forwhichtheb-functiondoes not change on each strata (see e.g. [8] for an algorithmic proof of this fact). In thedefinitions(2)and(3)forb-function,ifwereplace bythepolynomialring O k[x], by the Weyl algebra D, and by a holonomic D-module M = D/I, D M thenweobtaintheglobalb-functionforf andu. Itistheleastcommonmultiple of b-functions at every point. Theorem 3.1 Let u be the residue class of 1 in / I, and let b(s) be the b- D D function for f and u at a point p Cn where f(p)= 0. Assume that I admits ∈ an analytic solution of the form gfr around p, where r C, g is a holomorphic ∈ function on a neighborhood of p, and g(p)=0. Then s+r+1 divides b(s). 6 Proof: Let an and an berespectivelythesheafofanalyticdifferentialopera- D O torsandthe sheafofholomorphicfunctions onCn. We maydefine the analytic b-function by replacing by an, by an, and by a an-module an in O O D D M D M the definitions (2)and(3). Since theb-functionis ananalytic invariantandthe analytic and the algebraic b-functions coincide (see e.g. [8, 8]), we may work § in the analytic category. We do this to consider solutions gfr where g is holo- morphicatp. Ifweonlywishtoconsidersolutionsgfr whereg isapolynomial, then we may work in the algebraic category. In general, given a map of left an-modules φ : an an and a section D M1 →M2 u of an, the b-function for fsu at a point p is divisible by the b-function for M1 fsφ(u) at p. We apply this basic fact to the following map ϕ. Let Jan be the annihilating ideal of gfr in an. Since Jan Ian := anI and g(p) = 0, we D ⊇ D 6 have a left an-homomorphism D ϕ: an/Ian angfr = anfr ֒ an[f−1]fr D −→D D →O which sends u to gfr. This map extends to a left an[s]-homomorphism D 1 ϕ: an[f−1,s]fs Oan an/Ian ⊗ O ⊗ D an[f−1,s]fs Oan an[f−1]fr = an[f−1,s]fs+r −→O ⊗ O O whichsendsfs u togfs+r. By the definitionofb(s), there exists agermP(s) ⊗ of [s] at p such that D (P(s)f b(s))(fs u)=0. − ⊗ 5 Since1 ϕisaleft an-homomorphism,applyingittotheaboveequationgives ⊗ D the equation (P(s)f b(s))(gfs+r)=0, or in other words, − g−1P(s)gfs+r+1 =b(s)fs+r. Thus,weseethattheBernstein-Satopolynomialb (s)off atpdividesb(s r). f − Note that s+1 divides b (s) since f(p) = 0 (cf. [6]). In conclusion, we have f proved that s+1 divides b(s r). This completes the proof. [] − By virtue of the above theorem, we can obtain upper bounds by computing the b-function for fsu at a smooth point of each irreducible component of the singular locus of I. From now on, let us also take f k[x] to be a square-free ∈ polynomial defining the codimension one component of the singular locus, and let f =f f be its irreducible decomposition in k[x]. 1 m ··· Theorem 3.2 Let b (s) be the b-function for fsu at a generic point of f =0. i i i Denote by r the maximum integer root of b (s)=0. Then any rational solution i i (if any) to I can be written in the form gf−r1−1 f−rm−1 with a polynomial 1 ··· m g C[x]. Ifsomeb (s)hasnointegralroot,thenthereexistnorationalsolutions i ∈ to I other than zero. Proof: AnarbitraryrationalsolutiontoM iswrittenintheformgf−ν1 f−νm 1 ··· m with integers ν ,...,ν and g C[x]. Since the space of the rational solutions 1 m ∈ withcoefficientsinCisspannedbythosewithcoefficientsink,wemayassume g k[x], and f and g are relatively prime in k[x]. Let p be a generic point of ∈ f = 0. We may assume that f is smooth at p, g(p) = 0, and f (p) = 0 for i i j 6 6 j =i. It follows from Theorem3.1that b (ν 1)=0. This implies ν r +1. i i i i 6 − ≤ [] Since b-functions divide the global b-function, an upper bound can also be obtained from the global b-function. Corollary 3.3 Let b (s) be the global b-function for fsu, and denote by r the i i i maximum integer root of b (s)=0. Then any rational solution (if any) to I can i be written in the form gf−r1−1 f−rm−1 with a polynomial g C[x]. 1 ··· m ∈ We mention the corollarysince the algorithmto compute globalb-functions is simpler than the algorithm to compute b-functions. However, the b-function offers finer information. For instance, the well-known example f = x2 +y2+ z2+w2 hasBernstein-Satopolynomial(s+1)(s+2)comingfromthefunctional equation, 1(∂2 +∂2 +∂2 +∂2) fs+1 = (s+1)(s+2)fs. Now consider the 4 x y z w · module M =D f−1 and let u be the section of f−1. The global b-function for · fsu is s(s+1), and hence Corollary3.3 implies that rationalsolutions of M all have the form gf−1 or gf0, where g is a polynomial not divisible by f. On the other hand, the Bernstein-Sato polynomial of f at any nonsingular point p of f =0 (i.e. except for the origin) is s+1. It follows that the b-function for fsu equals s at the generic point of f = 0 and hence Theorem 3.2 implies that all rational solutions actually have the form gf−1. 6 An algorithm to compute the b-function and the global b-function for fsu wasfirstgivenin[8]basedupontensorproductcomputation,whichisslowand memoryintensive. Shortlythereafter,Waltherintroducedin[16]amoreefficient methodtocomputetheglobalb-functionforfsu. Bothmethodsgivetheglobal b-functionexactly,undertheconditionthatIisf-saturated. Otherwise,wegeta multipleoftheglobalb-function. Similarly,themethodof[8]givestheb-function exactly if I is f-saturated and additionally a certain primary decomposition in C[x] is known. If primary decomposition is only available in k[x], we againget a multiple of the b-function. Let us now describe an algorithm to compute the b-function for fsu at a generic point of f = 0 by combining the method of [16] and the primary decomposition as was used in [8]. Algorithm 3.4 (Computing an upper bound of the b-function at a generic point) Input: afinitesetG ofgeneratorsofaholonomicD-idealI andanirreducible 0 polynomial f k[x]. Output: b′(s∈) k[s], which is a multiple of the b-function b(s) for fsu at a ∈ generic point of f =0, where u is the residue class of 1 in D/I. 1. Introducing a new variable t, put ϑ = ∂ +(∂f/∂x )∂ . Let I be the i i i t left ideal of D , the Weyl algebra on the variables x ,...,x ,t, that is n+1 1 n e generated by P(x,ϑ ,...,ϑ ) P(x,∂ ,...,∂ ) G t f(x) . 1 n 1 n 0 { | ∈ }∪{ − } 2. Let G be a finite set of generators of the left ideal in (I) 1 (−1,0,...,0;1,0,...,0) of D . Here, 1 is the weight for t and 1 is the weight for ∂ . n+1 − t e 3. Rewrite each element P of G in the form 1 P =∂µP′(t∂ ,x,∂ ,...,∂ ) or P =tµP′(t∂ ,x,∂ ,...,∂ ) t t 1 n t 1 n with a non-negative integer µ, and define ψ(P) by, ψ(P):=tµ∂µP′ =t∂ (t∂ µ+1)P′(t∂ ,x,∂ ,...,∂ ) t t··· t− t 1 n or ψ(P):=∂µtµP′ =(t∂ +1) (t∂ +µ)P′(t∂ ,x,∂ ,...,∂ ). t t ··· t t 1 n Put G := ψ(P)( s 1,x,∂ ,...,∂ ) P G . 2 1 n 1 { − − | ∈ } 4. Compute the elimination ideal J := k[s,x] D[s]G . (The global b- 2 ∩ function can be obtained at this stage by computing the monic generator of the ideal J k[s].) ∩ 5. Compute a primary decomposition of J in k[s,x] as J =Q Q . 1 ν ∩···∩ 7 6. For each i=1,...,ν, compute Q :=Q k[x], which is a primary ideal ix i ∩ of k[x]. 7. Let b′(s) be the monic generator of the ideal Q k[s] Q k[x]f i ix { ∩ | ⊂ } \ p ofk[s]. (Note that√Q k[x]f implies that√Q equalsk[x]f or 0 .) ix ix ⊂ { } Theorem 3.5 In the above algorithm, the polynomial b′(s) is precisely the b- function for fsu at a generic point of f =0 if I is f-saturated(i.e., I :f∞ =I) and each C[s,x]Q remains primary in C[s,x]. Otherwise, the polynomial b′(s) i is a multiple of the b-function for fsu at a generic point of f =0. Proof: Using essentially the same method as the proof of Lemma 4.1 in [16], we can prove that I is precisely the annihilator ideal for δ(t f(x)) u in − ⊗ e M :=(Dn+1δ(t f(x))) C[x]D/I, − ⊗ whereδ(t f(x))denotfestheresidueclassof(t f(x))−1ink[x,(t f(x))−1]/k[x]. − − − Let b (s) be the indicial polynomial for δ(t f(x)) u along t = 0 at a point t − ⊗ (0,p) with f(p) = 0. Then by Theorem 6.14 of [8], the b-function b(s) for fsu at p divides out, and if I is f-saturated, coincides with b ( s 1). t − − It follows from the definition that b ( s 1) is a generator of the ideal t − − [s]J C[s] of C[s], where denotes the stalk of at p. If C[s,x]Q are p p i O ∩ O O primary in C[s,x], b′(s) generates the above ideal in view of Theorem 4.7 of [8] (cf. also Lemma 4.4 of [9]). In general, although Q is primary in k[s,x], i the extension C[s,x]Q is no longer primary in C[s,x] and admits a primary i decomposition C[s,x]Q =Q Q . i i1∩···∩ iµi In this case, b′(s) is the least common multiple of the generators of the ideals [s]Q C[s]for j =1,...,µ ,while b ( s 1)is the generatorof [s]Q p ij i t p ij O ∩ − − O ∩ C[s] for some j (such that p belongs to the zero set of Q C[x] which is also ij ∩ the zero set of a factor of f). This completes the proof. [] Remark 3.6 In the notationin the aboveproof, the linear factors of b′(s) and those ofb ( s 1) coincide. Inparticularthe set ofinteger rootsof b′(s)=0 is t − − the same as that of b ( s 1)= 0. In fact, this follows from the fact that the t − − linear factors of b′(s) in k[s] are invariant under the action of the Galois group of k over k. Remark 3.7 I isf-saturatedifandonlyifthe 1-thcohomologygroupofthe − restriction of M in the proof of Proposition 3.5 to t=0 vanishes (see Theorem 6.4 and Proposition 6.13 of [8]), which is computable by Algorithm 5.10 of [8] f or by Algorithm 5.4 of [9]. 8 Remark 3.8 The f-saturation of I, which is the ideal D[1] I D, may be f · ∩ computed by using the localization algorithm of [10] (if I is specializable along f) or by using the less efficient algorithm of [14] (if I is general). By replacing I with its f-saturation, we may then compute the local b-function exactly. However,sincesaturationisoftenanexpensivealgorithm,weavoidmakingthis replacement in practice. Oncewehavedeterminedtheintegersr ,...,r ofTheorem3.2,wecanuse 1 m Gröbnerdeformationstoobtaintherationalsolutions. Putk =r +1. Thenby i i virtue ofTheorem3.2,we haveonly to determine rationalsolutions of the form gf−k1 f−km for some polynomial g. This amounts to computing polynomial 1 ··· m solutions of some twisted ideal I of I. Namely, consider how f∂ acts (k1,...,km) i on an element gf−k1 f−km: 1 ··· m m ∂g f ∂f f∂i•(gf1−k1···fm−km)=f∂x − kjf ∂xjgf1−k1···fm−km. i Xj=1 j i   In other words, f∂ acts on the numerator g as the differential operator i m f ∂f j L =f∂ k . (4) i i j − f ∂x Xj=1 j i Thus, if we multiply a set of generators g ,...,g of I by sufficiently high 1 m { } powers fmi such that fmigi k x1,...,xn,f∂1,...,f∂n , and if I′ is the ideal ∈ h i obtained from fm1g1,...,fmrgr by substituting Li for f∂i, then the rational { } solutions of I have the polynomial solutions of I′ as numerators. Thus, it only remainsto computethe polynomialsolutionsofI′. However,sinceI′ mightnot be holonomic, we cannot apply Algorithm 2.4 just yet. HenceletusdefineI :=k(x) ∂ I′ D,whichistheWeylclosureof (k1,...,km) h i· ∩ I′andwhosepolynomialsolutionsarethesameasI. TheadvantageofI (k1,...,km) is that it is indeed holonomic, which follows from a theorem of Kashiwara. Namely, note that since I is of finite rank, I′ remains of finite rank because an element in I of the form (g (x)∂Ni + lower order elements) will be sent to an i i elementinI′oftheform(fMigi(x)∂iNi+ lower order elements). Nowleth(x)be any polynomial vanishing on the singular locus of I′. Then the non-holonomic locus of I′ is contained inside the zero set of h(x) regarded as a function on the cotangentbundle, and a theorem due to Kashiwara[7] states that the ideal D[h−1] I′ D is holonomic. Furthermore, an argument in [14] shows that the · ∩ Weyl closure of I′ also equals D[h−1] I′ D. Summing up, we arrive at the · ∩ following algorithm. Algorithm 3.9 (Computing the rational solutions of a holonomic ideal) Input: generators of a holonomic D-ideal I. Output: A basis of the rational solutions h k(x) of I h=0. ∈ • 1. Computeapolynomialf definingthecodimension1componentofSing(I). 9 2. Compute the irreducible decomposition f =f f in k[x]. 1 m ··· 3. For each i = 1,...,m, compute the output b′(s) of Algorithm 3.4 with I and f as input. Let r be the maximum integer root of b′(s)=0 and put i i k = r +1. If b′(s) has no integral root for some i, then there exists no i i rational solution other than zero. 4. Compute the twisted ideal I as follows. First, form the ideal (k1,...,km) I′ described in the paragraphspreceding the algorithm. Second, compute anypolynomialh(x)vanishingonthesingularlocusofI′. Third,compute the localization (D/I′)[h−1] using the algorithm in [10]. Then the ideal I is the kernel of D D/I′ (D/I′)[h−1]. (k1,...,km) → → 5. Compute a basis g ,...,g of the polynomial solutions of I { 1 k} (k1,...,km) using Algorithm 2.4. 6. Output: {g1f1−k1···fm−km,...,gkf1−k1···fm−km}, a basis of the rational solutions of I. Example 3.10 Let I be the left ideal generated by L =θ (θ +θ ) x(θ +θ +3)(θ 1) 1 x x y x y x − − and L =θ (θ +θ ) y(θ +θ +3)(θ +1). 2 y x y x y y − TheAppellfunctionF (3, 1,1,1;x,y)isasolutionofthissystem. Thesingular 1 − locus of I is xy(x y)(1 x)(1 y) = 0. We can compute the local indicial − − − polynomial of u, the modulo class of 1 in D /I, along x = 0 directly by the 2 algorithm of [8, Section 4]: It is s(s 1) on (0,y) y = 0 , and s(s 1)2 at − { | 6 } − (0,0). In the same way, the indicial polynomial of u along y =0 is s(s+1) on (x,0) x=0 , and s(s+1)(s 1) at (0,0). { | 6 } − Nowletuscomputetheb-functionfor(1 y)su. Thelocalindicialpolynomial − of δ(t+y 1) u along t = 0 is s(s+3) at any point of t = 0. Hence the − ⊗ b-functionfor(1 y)sudivides(s+1)(s 2). Inthesameway,thelocalindicial − − polynomial of δ(t+x 1) u along t = 0 is s(s+1) at any point of t = 0. − ⊗ Finally,the indicialpolynomialofδ(t x+y) uis s(s 1)on (x,x) x=0 , − ⊗ − { | 6 } and s(s 1)(s 2) at (0,0). − − Therefore, we conclude that any rational solution to I, if it exists, can be written in the form g(x,y)y−1(1 x)−1(1 y)−3 with a polynomial g. Now we − − maycompute thetwistedidealI ,wheref =x,f =y,f =x y,f = (0,1,0,1,3) 1 2 3 4 − x 1,f =y 1,andf istheproduct. Multiplyingbyf2,wegettheexpressions, 5 − − f2L = (x2 x3)(f∂ )2+x((1 3x)f (1 x)y∂f (1 x)x∂f)(f∂ )+ 1 − x − − − ∂y − − ∂x x x(1 x)y(f∂ )(f∂ )+xyf(f∂ )+3xf2 y x y − f2L = (y2 y3)(f∂ )2+y((1 5y)f (1 y)x∂f (1 y)y∂f)(f∂ )+ 2 − y − − − ∂x − − ∂y y y(1 y)x(f∂ )(f∂ ) yxf(f∂ ) 3yf2, x y x − − − 10