Intersection homology D-Modules and Bernstein associated with a complete intersection Tristan Torrelli

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Tristan Torrelli. Intersection homology D-Modules and Bernstein polynomials associated with a com- plete intersection. 2007. ￿hal-00171039v1￿

HAL Id: hal-00171039 https://hal.archives-ouvertes.fr/hal-00171039v1 Preprint submitted on 11 Sep 2007 (v1), last revised 25 May 2008 (v2)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. hal-00171039, version 1 - 11 Sep 2007 00MteaisSbetClassification: Subject 2000 Keywords: lt nescin,BrsenSt ucinlequation functional Bernstein-Sato intersections, plete nioi,Pr ars,018Nc ee ,France. 2, Cedex Nice 06108 Valrose, Parc Antipolis, ooopi ucin on functions holomorphic Abstract ooopi offiins tapoint a At coefficients. holomorphic Y cohomology D evresheaf perverse onie with coincides ino h spaces the of tion D oeby note Y [5) h tde h nlgu rbe ihatopologica holonomic a Kashiw regular of with the correspondence [17]), problem ([11], Riemann-Hilbert analogous the from the wor Indeed, a studies supplements who This ([15]), equations. functional Bernstein ing Let Introduction 1 equations. functional Bernstein-Sato rlnk-ahwr [].Ti stesmallest the is This ([5]). Brylinski-Kashiwara eto complex section hc onie with coincides which ainlhmlg sphere homology rational seuvln otefloigone: following the to equivalent is X,x X Let . 1 ⊂ nescinhomology Intersection Mdl fByisiKsiaa egv eea leri c algebraic an here give We Brylinski-Kashiwara. of -Module aua rbe st hrceietesubspaces the characterize to is problem natural A ie lsdsubspace closed a Given aoaor ..Dedn´,URd NS62,Universit´e 6621, CNRS du UMR Dieudonn´e, J.A. Laboratoire X ihtering the with ) X eacmlxaayi aiodo dimension of manifold analytic complex a be fpr codimension pure of L H nescinholomology intersection ( [ p Y ,X Y, Let . ] H ( O C [ p Y soitdwt opeeintersection complete a with associated H ) X ] Y IC ( X h ha flclagbacchmlg ihspotin support with cohomology algebraic local of sheaf the ) [ ⊂ p O [ Y Y p ] Y [] 9,[6)weeas where [16]) [9], ([7], ] eacmlxaayi aiod ie lsdsubspace closed a Given manifold. analytic complex a be X • ( O H O uhthat such ,and ), H [].B hswy hscondition this way, this By ([5]). [ X p Y = [ p Y .W rv eeta tmyb oelclyus- locally done be may it that here prove We ). ] ( ] C . O ( X O { X L x rsa Torrelli Tristan X eteitreto homology intersection the be ) and p 1 D Y ( ttegnrcpit of points generic the at ) x , . . . , ,X Y, L D ≥ Mdl n enti polynomials Bernstein and -Module ( mdls oa leri ooooygop com- group, cohomology algebraic local -modules, ⊂ ,X Y, D ,w osdrtesefo oa algebraic local of sheaf the consider we 1, D X ) h elln of link real the 24,3C8 24,3C5 14B05. 32C25, 32C40, 32C38, 32S40, X X x n ⊂ 1 h ha fdffrniloeaoswith operators differential of sheaf the onie with coincides ) -Module } ∈ fpr codimension pure of H (resp. X [ p Y eietf h stalk the identify we , L ] ( O ( ,X Y, E-mail: s. X D 1 H D h nescinhomology intersection the ) [ = p Y X orsod oteinter- the to corresponds ) ] n ( sboueof -submodule Y Oh O ≥ tristan X tapoint a at H ∂/∂x Y 2, L orsod othe to corresponds ) Y [ p Y ( O uhthat such ,X Y, ] [] [3]). ([5], ( [email protected] 1 O X ∂/∂x , . . . , fD Massey D. of k p X D eNc Sophia- Nice de etesefof sheaf the be ara-Mebkhout = ) ,i em of terms in ), ≥ X viewpoint. l haracteriza- Mdl of -Module O x ,w de- we 1, H H X,x ∈ L [ [ p p Y Y ( Y ] ] (resp. ,X Y, n ( ( O O i sa is ). X X ) ) ) First of all, we have an explicit local description of L(Y,X). This comes from the following result, due to D. Barlet and M. Kashiwara.

Y p p Theorem 1.1 ([3]) The fundamental class CX ∈ H[Y ](OX ) ⊗ ΩX of Y in p X belongs to L(Y,X) ⊗ ΩX

Y For more details about CX , see [1]. In particular, if h is an analytic mor- p phism (h1,...,hp):(X, x) → (C , 0) which defines the complete intersection p (Y, x) - reduced or not -, then the inclusion L(Y,X)x ⊂ H[Y ](OX )x may be identified with: .

mk1,...,kp (h) O[1/h1 ··· hp] Lh = D· ⊂ Rh = p h ··· hp ˇ k <

Theorem 1.2 Let Y ⊂ X be a hypersurface and h ∈ OX,x denote a local equation of Y at a point x ∈ Y . The following conditions are equivalent:

p 1. L(Y,X)x coincides with H[Y ](OX )x. 2. The reduced Bernstein of h has no root.

3. 1 is not an eigenvalue of the monodromy acting on the reduced coho- mology of the fibers of the Milnor fibrations of h around the singular points of Y , close enough to x.

Let us recall that the Bernstein polynomial bf (s) of a nonzero germ f ∈ O is the monic generator of the ideal of the polynomials b(s) ∈ C[s] such that:

b(s)f s = P (s) · f s+1 in O[1/f,s]f s, where P (s) ∈ D[s] = D ⊗ C[s]. The existence of such a nontrivial equation was proved by M. Kashiwara ([8]). When f is not a unit, it is easy to check that −1 isa rootof bf (s). The quotient of bf (s)by(s+1) is the so-called reduced Bernstein polynomial of f, denoted ˜bf (s). Let us recall that their roots are rational negative numbers in ] − n, 0[ when f is reduced ([24], [18]).

2 2 2 Example 1.3 Let f = x1 + ··· + xn. It is easy to prove that bf (s) is equal to (s + 1)(s + n/2), by using the identity:

∂ 2 ∂ 2 + ··· + · f s+1 = (2s + 2)(2s + n)f s ∂x ∂x  1 n 

In particular, Rf coincides with Lf if and only if n is odd.

These polynomials are famous because of the link of their roots with the monodromy of the Milnor fibration associated with f. This was estab- lished by B. Malgrange [14] and M. Kashiwara [10]. In particular, the set −2iπα {e | α root of ˜bf (s)} coincides with the one of the eigenvalues of the mon- odromy acting on the Grothendieck-Deligne vanishing cycle sheaf φf CX(x) (where X(x) ⊂ X is a sufficiently small neighborhood of x). Thus the equiv- alence 2 ⇔ 3 is an easy consequence of this deep fact (for more details, see [18]).

Remark 1.4 In [2], D. Barlet gives a characterization of 3 in terms of the λ meromorphic continuation of the current X(x) f .

In the case of hypersurfaces, it is wellR known that condition 1 requires a strong kind of irreducibility. This may be refinded in terms of Bernstein polynomial.

Proposition 1.5 Let f ∈ O be a nonzero germ such that f(0) = 0. Assume that f is reducible at a point x ∈ f −1{0} close enough to the origin. Then −1 is a root of the reduced Bernstein polynomial of f.

2 2 2 −1 Example 1.6 If f = x1 +x3x2, then (s+1) divides bf (s) because f {0} ⊂ 3 2 C is reducible at any (0, 0,λ), λ =6 0 (indeed, we have: bf (s)=(s +1) (s + 3/2)).

What may be done in higher codimensions ? If f ∈ O is such that (h, f) defines a complete intersection, we can consider the Bernstein polynomial bf (δh,s) of f associated with δh ∈ Rh. Indeed, we again have nontrivial functional equations: s s+1 b(s)δhf = P (s) · δhf with P (s) ∈D[s] (see part 2). This polynomial bf (δh,s) is again a multiple of (s + 1), and we can define a reduced Bernstein polynomial ˜bf (δh,s) as above. Meanwhile, in order to generalize Theorem 1.2, the suitable Bernstein polynomial is neither ˜bf (δh,s) nor bf (δh,s), but a third one trapped between these two.

3 n p+1 Notation 1.7 Given a morphism (h, f)=(h1,...,hp, f):(C , 0) → (C , 0) ′ defining a complete intersection, we denote by bf (h, s) the monic generator of the ideal of polynomials b(s) ∈ C[s] such that:

s s b(s)δhf ∈ D[s](Jh,f , f)δhf . (1)

′ Lemma 1.8 The polynomial bf (h, s) divides bf (δh,s), and bf (δh,s) divides ′ (s + 1)bf (h, s).

The second relation uses the identities:

p+1 s p+i+1 ∂ s+1 (s + 1)mk1,...,kp+1 (h, f)δhf = (−1) mk1,...,kˇi,...,kp+1 (h) ·δhf ∂xk " i=1 i # X ∆h k1,...,kp+1

| {z h } for 1 ≤ k1 < ··· < kp+1 ≤ n, where the vector field ∆k1,...,kp+1 annihilates ′ ˜ δh. In particular, bf (h, s) coincides with bf (δh,s) when −1 is not a root of ′ bf (h, s). But it is not always true (see part 3). We point out some facts about this polynomial in part 3.

Theorem 1.9 Let Y ⊂ X be a closed subspace of pure codimension p +1 ≥ p+1 2, and x ∈ Y . Let (h, f)=(h1,...,hp, f):(X, x) → (C , 0) be an analytic morphism defining (Y, x) and such that Dδh = Rh. The following conditions are equivalent:

p 1. L(Y,X)x coincides with H[Y ](OX )x.

′ 2. The polynomial bf (h, s) has no strictly negative integral root.

The proof of Theorems 1.2 & 1.9 is given in part 4. It is based on a natural generalization of a classical result due to M. Kashiwara which links α the roots of bf (s) to some generators of O[1/f]f , α ∈ C (see Proposition 4.2). The last part is devoted to remarks and comments about Theorem 1.9. Aknowledgements. This research has been supported by a Marie Curie Fellowship of the European Community (programme FP5, contract HPMD- CT-2001-00097). The author is very grateful to the Departamento de Algebra,´ Geometr´ıa y Topolog´ıa (Universidad de Valladolid) for hospitality during the fellowship, and to the Lehrstuhl VI f¨ur Mathematik (Universit¨at Mannheim) for hospitality in November 2005.

4 2 Bernstein polynomials associated with a sec- tion of an holonomic D-module

In this paragraph, we recall some results about Bernstein polynomials asso- ciated with a section of an holonomic D-module.

Given a nonzero germ f ∈ OX,x ∼= O and a local section m ∈ Mx of a holonomic DX -Module without f-torsion, M. Kashiwara proved that there exists a functional equation:

b(s)mf s = P (s) · mf s+1 in (Dm) ⊗ O[1/f,s]f s, where P (s) ∈ D[s] = D ⊗ C[s] and b(s) ∈ C[s] are nonzero ([9]). The Bernstein polynomial of f associated with m, denoted by bf (m, s), is the monic generator of the ideal of polynomials b(s) ∈ C[s] which satisfies such an equation. When f is not a unit, it is easy to check that if m ∈Mx − fMx, then −1 is a root of bf (m, s). Of course, if M = OX and m = 1, this is the classical notion recalled in the introduction.

Let us recall that when M is a regular holonomic DX -Module, the roots of the polynomials bf (m, s) are closely linked to the eigenvalues of the mon- odromy of the perverse sheaf ψf (Sol(M)) around x, the Grothendieck-Deligne nearby cycle sheaf, see [12] for example. Here Sol(M) is the complex RHomDX (M, OX ) of holomorphic solutions of M, and the relation is similar to the one given in the introduction (since Sol(OX ) ∼= CX ). This comes from the algebraic con- struction of vanishing cycles, using Malgrange-Kashiwara V -filtration [14], [10]. Now, if Y ⊂ X is a subspace of pure codimension p, then the regular p holonomic DX -Module H[Y ](OX ) corresponds to CY [p]. Thus the roots of the p polynomials bf (δ, s), δ ∈ H[Y ](OX )x, are linked to the monodromy associated with f :(Y, 0) → (C, 0). For more results about these polynomials, see [19].

′ ˜ 3 The polynomials bf (h, s) and bf (δh, s)

′ Let us recall that bf (h, s) is always equal to one of the two polynomials bf (δh,s) and ˜bf (δh,s). In this paragraph, we point out some facts about these Bernstein polynomials associated with an analytic morphism (h, f)= n p+1 (h1,...,hp, f):(C , 0) → (C , 0) defining a complete intersection.

5 ′ First we have a closed formula for bf (h, s) when h and (h, f) define weighted-homogeneous isolated complete intersection singularities.

Proposition 3.1 ([20]) Let f, h1,...,hp ∈ C[x1,...,xn], p < n, be some ∗+ weighted-homogeneous of degree 1, ρ1,...,ρp ∈ Q for a system of weights ∗+ n α = (α1,...,αn) ∈ (Q ) . Assume that the morphisms h = (h1,...,hp) and (h, f) define two germs of isolated complete intersection singularities. ′ Then the polynomial bf (h, s) is equal to:

(s + |α| − ρh + q) q∈Π Y n p + where |α| = i=1 αi, ρh = i=1 ρi and Π ⊂ Q is the set of the weights of the elements of a weighted-homogeneous of O/(f, h1,...,hp)O + Jh,f . P P ′ When h is not reduced, the determination of bf (h, s) is more difficult, even if (h, f) is a homogeneous morphism.

2 2 ℓ Example 3.2 Let p = 1, f = x1 and h = (x1 + ··· + xn) with ℓ ≥ 1. By ˜ using a formula given in [22], Remark 4.12, the polynomial bx1 (δh,s) is equal ∗ ′ to (s + n − 2ℓ) for any ℓ ∈ N . For ℓ ≥ n/2, let us determine bx1 (h, s) with the help of Theorem 1.9. From Example 1.3, we have Rh = Dδh if ℓ ≥ n/2, and Lh,x1 = Rh,x1 if and only if n is even. ′ If n ≤ 2ℓ is odd, bx1 (h, s) must coincide with bx1 (δh,s)=(s+1)(s+n−2ℓ) because of Theorem 1.9 (since Lh,x1 =6 Rh,x1 ). On the other hand, if n ≤ 2ℓ ′ ˜ is even, we have bx1 (δh,s)=(s + n − 2ℓ)= bx1 (δh,s) by the same arguments. Let us refind this last fact by a direct calculus. ˜ ′ As bx1 (δh,s)=(s + n − 2ℓ) divides bx1 (h, s), we just have to check that ˜ this polynomial (s+n−2ℓ) provides a functional equation for bx1 (δh,s) when n is even. First, we observe that

n s ∂ s (s + n − 2ℓ)δhx1 = xi · δhx1 ∂xi " i=1 # X

If ℓ = 1, we get the result (since Jh,x1 =(x2,...,xn)O in that case). Now we s s assume that ℓ ≥ 2. Let us prove that xiδhx1 belongs to D(Jh,x1, x1)δhx1 for 2 2 2 ≤ i ≤ n. We denote by g the polynomial x1 +···+xn and, for 0 ≤ j ≤ ℓ−1, s s j s by Nj ⊂D(Jh,x1 , x1)δhx1 the submodule generated by x1δhx1, x2g δhx1, ..., j s ℓ s xng δhx1. In particular, h = g , Nℓ−1 = D(Jh,x1 , x1)δhx1 and Nj+1 ⊂Nj for 1 ≤ j ≤ ℓ − 2. To conclude, we have to check that N0 = Nℓ−1.

6 By a direct computation, we obtain the identity:

n ∂ j−1 2 ∂ j s j−1 s 2(j − ℓ)g x1 + xkg ·δhx1 =(j−ℓ)(n+2(j−ℓ)−1)xig δhx1 ∂xi ∂xk " k # X=2 j−1 s for 2 ≤ i ≤ n, j > 0. As n is even, we deduce that xig δhx1 belongs to Nj for 2 ≤ i ≤ n. In other words, Nj−1 = Nj for 1 ≤ j ≤ ℓ − 1; thus N0 = Nℓ−1, as it was expected.

′ ˜ As the polynomial bf (h, s) plays the rule of bf (s) in Theorem 1.9, a natural ′ ˜ question is to compare the polynomials bf (h, s) and bf (δh,s). Of course, when ′ ′ ˜ (s +1) is not a factor of bf (h, s), then bf (h, s) must coincide with bf (δh,s); from Theorem 1.9, this sufficient condition is satisfied when Dδh = Rh and Rh,f = Lh,f . But in general, all the cases are possible (see Example 3.2). Let us study this problem when (h, f) define isolated complete intersection singularity. In that case, let us consider the short exact sequence:

s s D[s]δhf ։ D[s]δhf K ֒→ s (s + 1) s+1 → 0→ 0 D[s](Jh,f , f)δhf D[s]δhf where the three D-modules are supported by the origin. Thus the polynomial ′ ˜ ˜ bf (h, s) is equal to l.c.m(s+1, bf (δh,s)) if K= 6 0 and it coincides with bf (δh,s) if not. Remark that K is not very explicit, since there does not exist a general Bernstein functional equation which defines ˜bf (δh,s) - contrarily to ˜bf (s), see part 4. In [21], [22], we have investigated some contexts where such a functional equation may be given. In particular, this may be done when the following condition is satisfied:

A(δh) : The ideal AnnD δh of operators annihilating δh is generated by AnnO δh and operators Q1,...,Qw ∈D of order 1.

s+1 s Indeed, because of the relations: Qi · δhf =(s + 1)[Qi, f]δhf , 1 ≤ i ≤ w, we have the following isomorphism:

D[s]δ f s ∼ D[s]δ f s h −→= (s + 1) h s s+1 D[s](Jh,f , f)δhf D[s]δhf where Jh,f ⊂ O is generatede by the commutators [Qi, f] ∈ O, 1 ≤ i ≤ w. Thus ˜bf (δh,s) may also be defined using the functional equation: e s s b(s)δhf ∈D[s](Jh,f , f)δhf

s s and K = D[s](Jh,f , f)δhf /D[s](Jh,f , f)eδhf . For more details about this condition A(δh), see [23]. e 7 4 The proofs

Let us recall that ˜bh(s) may be defined as the unitary nonzero polynomial b(s) ∈ C[s] of smallest degree such that:

n s s+1 ′ s b(s)h = P (s) · h + Pi(s)hxi h (2) i=1 X where P (s),P1(s),...,Pn(s) ∈D[s] (see [13]).

s+1 ′ ′ s Remark 4.1 In fact, one can prove that h ∈ D[s](hx1 ,...,hxn )h i.e. h ′ ′ s belongs to the ideal I = D[s](hx1 ,...,hxn ) + AnnD[s]h . This requires some ′ computations like in [22] 2.1., using the fact that: h∂xi − shxi ∈ I,1 ≤ i ≤ n.

Proof of Proposition 1.5. By semi-continuity of the Bernstein polynomial, it is enough to prove the assertion for a reducible germ f. Let us write f = f1f2 where f1, f2 ∈ O have no common factor. Assume that −1 is not a root of ˜bf (s). Then, by fixing s = −1 in (2), we get:

n 1 f ′ ∈ D xi + O ⊂ O[1/f ]+ O[1/f ] f f 1 2 i=1 X ′ ′ ′ since fxi /f = f1,xi /f1 + f2,xi /f2, 1 ≤ i ≤ n. But this is absurd since 1/f1f2 defines a nonzero element of O[1/f1f2]/O[1/f1]+ O[1/f2] under our assump- tion on f1, f2. Thus −1 is a root of ˜bf (s).  Up to notational changes2, the proofs of Theorem 1.2 and 1.9 are identical. They are based on the following result.

Proposition 4.2 Let f ∈ O be a nonzero germ such that f(0) = 0. Let m be a section of a holonomic D-module M without f-torsion, and ℓ ∈ N∗. The following conditions are equivalent :

1. The smallest integral root of bf (m, s) is strictly greater than −ℓ − 1.

2. The D-module (Dm)[1/f] is generated by mf −ℓ. . 3. The D-module (Dm)[1/f]/Dm is generated by mf −ℓ. 2 ′ ˜ Change δh by 1, Rh by O, bf (h,s) by bh(s), equation (1) by equation (2). . .

8 4. The following D-linear morphism is an isomorphism :

D[s]mf s −→ Dm[1/f] (s + ℓ)D[s]mf s . P (s)mf s 7→ P (−ℓ)mf −ℓ

This is a direct generalization of a well known result due to M. Kashiwara and E. Bj¨ork for m =1 ∈ O = M ([8] Proposition 6.2, [4] Proposition 6.1.18, 6.3.15 & 6.3.16). For more details, see [19]. Let us prove Theorem 1.9.

′ Proof of Theorem 1.9. If bf (h, s) has no strictly negative integral root, then Dδh,f = Rh,f (Lemma 1.8 and Proposition 4.2, using that Dδh = Rh), and we just have to remark that δh,f belongs to Lh,f when −1 is not a ′ root of bf (h, s). Indeed, by fixing s = −1 in (1), we get: δh ⊗ 1/f ∈ n ∼ 1≤k1<···

′ ′ by (1), and then bf (h, s) divides bf (h, s)/(s + 1), which is absurd. Therefore ′  −1 is not a root of bf (h, s), and this ends the proof.

Remark 4.3 Under the assumption Dδh,f = Rh,f , we show that if δh,f be- ′ longs to Lh,f then −1 is not a root of bf (h, s). As the reverse relation is obvious, a natural question is to know if this assumption is necessary. In terms of reduced Bernstein polynomial, does the condition: −1 is not a root of ˜bh(s) caracterize the membership of δh in Lh ?

9 5 Some remarks

Let us point out some facts about Theorem 1.9:

- The assumption Dδh = Rh is necessary. This appears clearly in the following examples.

2 2 Example 5.1 Let p = 1 and h = x1 + ··· + x4. As bh(s)=(s + 1)(s + 2), ′ we have Dδh =6 Rh (Proposition 4.2). If f1 = x1, then bf1 (h, s)=(s + 2) by using Proposition 3.1 where as Lh,f1 = Rh,f1 (Example 1.3, or because ′ Dδf1 = Rf1 and bf1 (h, s)=(s +3/2)). ′ Now if we take f2 = x5, we have Lh,f2 =6 Rh,f2 and bf2 (h, s) = 1 since:

. 4 . 2 s ∂ 1 s 2 2 x5 = xi · 2 2 x5 . x + ··· + x ∂xi x + ··· + x 1 4 " i=1 # 1 4 X

- If p = 1, this condition Dδh = Rh just means that the only integral root of bh(s) is −1 (Proposition 4.2). By iterating this Proposition, one can obtain a criterion for this condition (see [21]).

- The condition Lh = Rh clearly implies Dδh = Rh, but it is not necessary; see Example 1.6 for instance.

- If (h, f)=(h1,...,hp, f) is such that (V (h1,...,hp, f), 0) = (Y, x), it ˙ ℓ1 ℓp ∗ is easy to see that D1/h1 ··· hp = Rh for ℓ1,...,ℓp ∈ N big enough, using Proposition 4.2. For example, one can take ℓ1 = ··· = ℓp = n − 1 since n−1 1/(h1 ··· hp) generates the D-module O[1/h1 ··· hp] (using the boundaries of the roots of the classical Bernstein polynomial). In the previous example, we have Dδh2 = Rh and we can check that ′ 2 ′ 2 bf1 (h ,s)= s and bf2 (h ,s)=(s + 1). This agrees with the theorem. - Contrarly to the classical Bernstein polynomial, it may happen that an ′ integral root of bf (h, s) is positive or zero (see Example 3.2, with f = x1, 2 2 ℓ h = (x1 + ··· + x4) and ℓ ≥ 2). In particular, 1 is an eigenvalue of the monodromy acting on φf Ch−1{0}. For that reason, we do not have here the analogue of condition 3, Theorem 1.2. - In [6], the authors introduce a notion of Bernstein polynomial for an arbitrary variety Z. In the case of hypersufaces, this polynomial bZ (s) co- incides with the classical Bernstein-Sato polynomial. But it does not seem to us that its integral roots are linked to the condition Lh,f = Rh,f . For 2 2 2 2 2 2 instance, if h = x1 + x2 + x3 and f = x4 + x5 + x6 then one can check that

10 ′ ˜ s bf (h, s)= b(f ,s)=(s+3/2); in particular Lh,f = Rh,f . Meanwhile, by using 6 [6], Theorem 5, we get bZ (s)=(s + 3)(s +5/2)(s +2) if Z = V (h, f) ⊂ C .

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