An effective method for computing Grothendieck point residue mappings
aa r X i v : . [ c s . S C ] N ov An effective method for computingGrothendieck point residue mappings
Shinichi Tajima
Graduate School of Science and Technology, Niigata University,8050, Ikarashi 2-no-cho, Nishi-ku Niigata, Japan
Katsusuke Nabeshima
Graduate School of Technology, Industrial and Social Sciences, TokushimaUniversity, Minamijosanjima-cho 2-1, Tokushima, Japan
Abstract
Grothendieck point residue is considered in the context of computational com-plex analysis. A new effective method is proposed for computing Grothendieckpoint residues mappings and residues. Basic ideas of our approach are the use ofGrothendieck local duality and a transformation law for local cohomology classes. Anew tool is devised for efficiency to solve the extended ideal membership problemsin local rings. The resulting algorithms are described with an example to illustratethem. An extension of the proposed method to parametric cases is also discussedas an application.
Key words:
Grothendieck local residues mapping, algebraic local cohomology,transformation law
The theory of Grothendieck residue and duality is a cornerstone of algebraicgeometry and complex analysis (Griffiths and Harris, 1978; Grothendieck, 1957; ⋆ This work has been partly supported by JSPS Grant-in-Aid for Scientific Research(C) (Nos 18K03214,18K03320).
Email addresses: [email protected] (Shinichi Tajima), [email protected] (Katsusuke Nabeshima).
19 November 2020 artshorne, 1966; Kunz, 2009). It has been used and applied in diverse prob-lems of several different fields of mathematics (Baum and Bott, 1972; Bykov et al.,1991; Cardinal and Mourrain, 1996; Dickenstein and Sessa, 1991; Griffiths,1976; Lehmann, 1991; O’Brian, 1975; Perotti, 1998; Suwa, 2005). In the globalsituation, methods for computing the total sum of Grothendieck residues havebeen extensively studied and applied by several authors (Bykov et al., 1991;Cattani et al., 1996; Kytmanov, 1988; Yushakov, 1984).The concept of Grothendieck local residue together with the local dualitytheory also play quite important roles in complex analysis, especially in sin-gularity theory (Brasselet et al., 2009; Cherveny, 2018; Corrˆea et al., 2016;Klehn, 2002; O’Brian, 1975; Suwa, 1988). Computing Grothendieck localresidues is therefore of fundamental importance. However, since the prob-lem is local in nature, it is difficult in general to compute Grothendieck localresidues (O’Brian, 1977). In fact, a direct use of the classical transformationlaw described in (Hartshorne, 1966) only gives algorithms which lack effi-ciency. Compared to the global situation, despite the importance, much lesswork has been done on algorithmic aspects of computing Grothendieck lo-cal residues (Elkadi and Mourrain, 2007; Mourrain, 1997; Ohara and Tajima,2019a,b; Tajima and Nakamura, 2005b). Grothendieck local residues with pa-rameters are useful in the study of singularity theory, for example, deforma-tions of singularity and unfoldings of holomorphic foliations (Kulikov, 1998;Saito, 1983; Varchenko, 1986). However, to the best of our knowledge, exist-ing algorithm of computing Grothendieck local residues are not designed tobe able to treat parametric cases.In this paper, we consider methods for computing Grothendieck point residuesfrom the point of view of complex analysis and singularity theory. We proposea new effective method for computing Grothendieck point residues mappingsand residues, which can be extended to treat parametric cases.Let X ⊂ C n be an open neighborhood of the origin O ∈ C n and let f ( z ) , f ( z ) ,. . . , f n ( z ) be n holomorphic functions defined on X , where z = ( z , z , . . . , z n ) ∈ X . Assume that their common locus in X is the origin O : { z ∈ X | f ( z ) = f ( z ) = · · · = f n ( z ) = 0 } = { O } .Then, for a given germ h ( z ) of holomorphic function at O , the Grothendieckpoint residue at the origin O , denoted byres { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! , of the differential form h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) can be expressed, or defined, as2he integral π √− ! n Z · · · Z γ ǫ h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) , where dz = dz ∧ dz ∧ · · · ∧ dz n , and where γ ǫ is a real n -dimensional cycle: γ ǫ = { z ∈ X | | f ( z ) | = | f ( z ) | = · · · = | f n ( z ) | = ǫ } , with 0 < ǫ ≪ . (See for instance, (Baum and Bott, 1972; Griffiths and Harris,1978; Tong, 1973)).Let h ( z ) −→ res { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! be the Grothendieck point residue mapping that assigns to a holomorphicfunction h ( z ) the value of the Grothendieck point residue. We show that, basedon the concept of local cohomology, the use of Grothendieck local duality and atransformation law for local cohomology classes given by J. Lipman (Lipman,1984) allows us to design an effective method for computing Grothendiecklocal residue mappings and another one for computing Grothendieck localresidues. Note that the classical transformation law on Grothendieck residueis of no avail for computing Grothendieck local residue mappings. Since wecompute Grothendieck local residue mappings, our method is applicable whenthe holomorphic function h ( z ) in the numerator is computable, that is thecase when the coefficients of the Taylor expansion of h ( z ) is computable. Thisis an advantage of our approach. We also show that the proposed methodcan be extended to treat parametric cases. This is another advantage of ourapproach.In Section 2, we recall the transformation law for local cohomology classes andGrothendieck local duality. In Section 3, we fix our notation and we brieflyrecall our basic tool, an algorithm for computing Grothendieck local duality.We devise, in the context of exact computation, a new tool which plays a keyrole in the resulting algorithm. In Section 4, we describe the resulting algo-rithm for computing Grothendieck point residue mappings and the algorithmfor computing Grothendieck point residues. In Section 5, as an application,we generalize the proposed method to treat parametric cases and we show, byusing an example, an algorithm for computing Grothendieck point residuesassociated to a µ -constant deformation of quasi homogeneous isolated hyper-surface singularities. 3 Local analytic residues
The concept of Grothendieck point residue was introduced by A. Grothendieckin terms of derived categories and local cohomology. In this section, we brieflyrecall some basics on transformation law for local cohomology classes andGrothendieck local duality.Let X ⊂ C n be an open neighborhood of the origin O ∈ C n . Let O X bethe sheaf on X of holomorphic functions, and Ω nX the sheaf of holomorphic n -forms. Let H n { O } ( O X ) (resp. H n { O } (Ω nX ) ) denote the local cohomology sup-ported at O of O X (resp. Ω nX ).Then, O X,O , the stalk at O of the sheaf O X , and the local cohomology H n { O } (Ω nX )are mutually dual as locally convex topological vector spaces (B˘anic˘a and St˘an˘a¸sil˘a,1974). The duality is given by the point residue pairing:res { O } ( ∗ , ∗ ) : O X,O × H n { O } (Ω nX ) −→ C Let F = [ f ( z ) , f ( z ) , . . . , f n ( z )] be an n -tuple of n holomorphic functionsdefined on X . Assume that their common locus { z ∈ X | f ( z ) = f ( z ) = · · · = f n ( z ) = 0 } in X is the origin O. Let I F denote the ideal in O X,O generated by f ( z ) , f ( z ) , . . . , f n ( z ) . Let ω F denote a local cohomology class ω F = dzf ( z ) f ( z ) · · · f n ( z ) in H n { O } (Ω nX ), where dz = dz ∧ dz ∧· · ·∧ dz n , and [ ] stands for Grothendiecksymbol (Hartshorne, 1966; Grothendieck, 1967). Residue theory says that, for h ( z ) in O X,O , one hasres { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! = res { O } ( h ( z ) , ω F ) . Since V ( I F ) ∩ X = { O } , there exists, for each i = 1 , , . . . , n, a positive integer m i such that z m i i ∈ I F . There exists an n -tuple of holomorphic functions a i, ( z ) , a i, ( z ) , . . . , a i,n ( z ) such that z m i i = a i, ( z ) f ( z ) + a i, ( z ) f ( z ) + · · · + a i,n ( z ) f n ( z ) , i = 1 , , . . . , n. Set A ( z ) = det( a i,j ( z )) ≤ i,j ≤ n .
4e have the following key lemma (Lipman, 1984).
Lemma 1 (Transformation law for local cohomology classes) . In H n { O } (Ω nX ) , the following formula holds. ω F = A ( z ) dzz m z m · · · z m n n . For the proof of the result above, we refer the reader to (Kunz, 2009; Lipman,1984). Note that the formula above implies the classical transformation lawres { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! = res { O } h ( z ) A ( z ) dzz m z m · · · z m n n ! for point residues described in (Hartshorne, 1966). See also (Baum and Bott,1972; Boyer and Hickel, 1997; Griffiths and Harris, 1978; Kytmanov, 1988). We define W F to be the set of local cohomology classes in H n { O } (Ω nX ) that arekilled by I F : W F = { ω ∈ H n { O } (Ω nX ) | f ( z ) ω = f ( z ) ω = · · · = f n ( z ) ω = 0 } . Then, according to Grothendieck local duality, the pairingres { O } ( ∗ , ∗ ) : O X,O /I F × W F −→ C induced by the residue mapping is non-degenerate (Altman and Kleiman, 1970;Grothendieck, 1957; Hartshorne, 1966; Lipman, 2002).Let ≻ − be a local term ordering on the local ring O X,O and let { z α | α ∈ Λ F } denote the monomial basis of the quotient space O X,O /I F with respect to thelocal term ordering ≻ − , where Λ F ⊂ N n is the set of exponents α of basismonomials z α .Let { ω α ∈ W F | α ∈ Λ F } denote the dual basis of { z α | α ∈ Λ F } with respectto the Grothendieck point residue. Then, we have(i) O X,O /I F ∼ = Span C { z α | α ∈ Λ F } , (ii) W F = Span C { ω α | α ∈ Λ F } , { O } ( z α , ω β ) = , α = β, , α = β, α, β ∈ Λ F . Since ω F satisfies f ( z ) ω F = f ( z ) ω F = · · · = f n ( z ) ω F = 0 , the local cohomol-ogy class ω F is in W F . Therefore ω F can be expressed as a linear combinationof the basis { ω α | α ∈ Λ F } . Assume that, for the moment, we have the following expression: ω F = X α ∈ Λ F b α ω α , b α ∈ C . Now let NF ≻ − ( h )( z ) = X α ∈ Λ F h α z α , h α ∈ C be the normal form of the given holomorphic function h ( z ) . Then, we havethe following.
Theorem 2. res { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! = X α ∈ Λ F h α b α Proof.
Since h − NF ≻ − ( h ) ∈ I F , we haveres { O } ( h ( z ) , ω F ) = res { O } (NF ≻ − ( h )( z ) , ω F ) . Therefore,res { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! = res { O } X α ∈ Λ F h α z α , X β ∈ Λ F b β ω β , which is equal to X α,β ∈ Λ F h α b β res { O } ( z α , ω β ) = X α ∈ Λ F h α b α . This completes the proof. 6
Tools
Let us consider a method for computing Grothendieck point residues in thecontext of symbolic computation. We start by recalling some basics on an algo-rithm for computing Grothendieck local duality given in (Tajima and Nakamura,2009; Tajima et al., 2009).Let K = Q be the field of rational numbers and let z = ( z , z , . . . , z n ) ∈ C n . Let H n [ O ] ( K [ z ]) denote the algebraic local cohomology defined to be H n [ O ] ( K [ z ]) = lim k →∞ Ext nK [ z ] ( K [ z ] / m k , Ω nX ) , where m is the maximal ideal m = h z , z , . . . , z n i in K [ z ] = K [ z , z , . . . , z n ] . We adopt the notation used in (Nabeshima and Tajima, 2015a,b, 2016a,b)to handle local cohomology classes. For instance, a polynomial P λ c λ ξ λ in K [ ξ ] = K [ ξ , ξ , . . . , ξ n ] represents the local cohomology class of the form X λ =( ℓ ,ℓ ,...,ℓ n ) c λ z ℓ +11 z ℓ +12 · · · z ℓ n +1 n . Note that a multiplication on ξ β by z α is z α ∗ ξ β = ξ β − α , β ≥ α. , otherwise.Let ≻ be a term ordering on K [ ξ ]. For a local cohomology class ψ = c α ξ α + X ξ α ≻ ξ γ c γ ξ γ , we call ξ α the head monomial of ψ , and α ∈ N n the head exponentof ψ .Let F = [ f ( z ) , f ( z ) , . . . , f n ( z )] be a list of n polynomials f , . . . , f n in K [ z ].We also assume as in the previous section that there exists an open neighbor-hood X of the origin O such that their common locus is the origin: { z ∈ X | f ( z ) = f ( z ) = · · · = f n ( z ) = 0 } = { O } . We set H F = { ψ ∈ H n [ O ] ( K [ z ]) | f ( z ) ∗ ψ = f ( z ) ∗ ψ = · · · = f n ( z ) ∗ ψ = 0 } . In (Nabeshima and Tajima, 2017; Tajima et al., 2009), an algorithm for com-puting bases of H F is introduced. Let Ψ F denote an output of the algorithm.Then, W F = Span C { ψdz | ψ ∈ Ψ F } F , a basis of the vector space H F , has thefollowing form: Ψ F = ψ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ α = ξ α + X ξ α ≻ ξ γ c γ ξ γ , α ∈ Λ F , where Λ F ⊂ N n is the set of the head exponents of local cohomology classesin Ψ F .Let L F denote the set of lower exponents of local cohomology classes in Ψ F : L F = γ ∈ N n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∃ ψ α = ξ α + X ξ α ≻ ξ γ c γ ξ γ ∈ Ψ F such that c γ = 0 . Set E F = Λ F ∪ L F and T F = { ξ λ | λ ∈ E F } . Now let ℓ F,i = max { ℓ | ξ ℓi ∈ T F } . Then Grothendieck local duality implies the following.
Lemma 3.
Set m i = ℓ F,i + 1 . Then z m i i ∈ I F holds, where I F is the ideal inthe local ring K { z } generated by f ( z ) , f ( z ) , . . . , f n ( z ) . Proof.
Since z m i i ∗ ψ α = 0 and ψ α ∈ Ψ F hold, we have z m i i ∗ ψ = 0 for ψ ∈ H F . It follows from the Grothendieck local duality that z m i i is in I F . Now let us consider the set of monomials M F in K { z } defined to be M F = { z α | α ∈ Λ F } . Let ≻ − denote the local term ordering on K { z } defined as theinverse ordering of ≻ . Then, M F constitutes a monomial basis of the quotient K { z } /I F with respect to the local term ordering ≻ − . Furthermore, we havethe following result (Tajima and Nakamura, 2005a,b, 2009).
Theorem 4.
Let Ψ F , M F be as above. Then, Ψ F is the dual basis of the basis M F with respect to Grothendieck local residue pairing. That is, for z α ∈ M F and for ψ β ∈ Ψ F , res { O } ( z α , ψ β dz ) = , α = β, , α = β, holds.Sketch of the proof . Since the algorithm outputs a reduced basis of H F , wehave Λ F ∩ L F = ∅ , which implies the result.8 .2 A key tool Let m i be an integer such that z m i i is in the ideal I F = ( f , f , . . . , f n ) in thelocal ring. Then there exist germs a i, ( z ) , a i, ( z ) , . . . , a i,n ( z ) of holomorphicfunctions such that z m i i = a i, ( z ) f ( z ) + a i, ( z ) f ( z ) + · · · + a i,n ( z ) f n ( z ) , i = 1 , , . . . , n. Theory of symbolic computation asserts that such n -tuple of holomorphic func-tions can be obtained by computing syzygies in the local ring K { z } . Whereas,since the cost of computation of syzygies in local rings is high, a direct useof the classical algorithm of computing syzygy is not appropriate in actualcomputations. In fact, it is difficult to obtain these holomorphic functions. Inprevious papers (Nabeshima and Tajima, 2016b), the authors of the presentpaper have proposed a new effective method to overcome this type of difficulty.We adopt the proposed method mentioned above and devise a new, muchmore efficient algorithm by improving the previous algorithm presented in(Nabeshima and Tajima, 2015b, 2016b). We start by recalling the main ideagiven in (Nabeshima and Tajima, 2016b). Let J F = h f ( z ) , f ( z ) , . . . , f n ( z ) i denote the ideal in the polynomial ring K [ z ] generated by f ( z ) , f ( z ) , . . . , f n ( z ) . Let J F,O be the primary component of J F whose associated prime is the max-imal ideal m = h z , z , . . . , z n i , and G Q a Gr¨obner basis of the ideal quotient Q = J F : J F,O ⊂ K [ z ] . Then there is in G Q a polynomial, say q ( z ) , such that q ( O ) = 0 . Now let r ( z ) ∈ J F,O . Then, since q ( z ) r ( z ) ∈ J F , there exists an n -tuple ofpolynomials p ( z ) , p ( z ) , . . . , p n ( z ) in K [ z ], such that q ( z ) r ( z ) = p ( z ) f ( z ) + p ( z ) f ( z ) + · · · + p n ( z ) f n ( z ) . Since, q ( O ) = 0 , we have a following expression in the local ring K { z } : r ( z ) = p ( z ) q ( z ) f ( z ) + p ( z ) q ( z ) f ( z ) + · · · + p n ( z ) q ( z ) f n ( z ) . Since I F = K { z } ⊗ J F,O and z m i i ∈ I F , z m i i ∈ J F,O holds. Therefore, theargument above can be applied to compute germs a i, ( z ) , a i, ( z ) , . . . , a i,n ( z ) ofholomorphic functions. Note also that, since J F,O = { p ( z ) ∈ K [ z ] | p ( z ) ∗ ψ α =0 , ψ α ∈ Ψ F } , the primary ideal J F,O can be computed by using Ψ F . Let G F = { g , g , . . . , g ν } be a Gr¨obner basis of J F . Let R F be a list of relationsbetween g j and F = [ f , f , . . . , f n ] : g j = r ,j f + r ,j f + · · · + r n,j f n , r i,j ∈ K [ z ] , i = 1 , , . . . , n , and j = 1 , , . . . , ν. Let S F be a Gr¨obnerbasis of the module of syzygies among F : s f + s f + · · · + s n f n = 0 , where s i ∈ K [ z ] , i = 1 , , . . . , n. Let q be a polynomial in G Q such that q ( O ) = 0 . Now we are ready to present a new tool.
Algorithm 1 . localexpressionInput : G F , R F , S F , q, r. Output : [ p , p , . . . , p n ] such that q ( z ) r ( z ) = p ( z ) f ( z ) + p ( z ) f ( z ) + · · · + p n ( z ) f n ( z ) . BEGIN step 1: divide qr by the Gr¨obner basis G F = { g , g , ..., g ν } : qr = e g + e g + · · · + e ν g ν ;step 2: rewrite the relation above by using R F : qr = (cid:16)P j r j, e j (cid:17) f + (cid:16)P j r j, e j (cid:17) f + · · · + (cid:16)P j r j, e j (cid:17) f n ;step 3: simplify the expression above by using S F : q ( z ) r ( z ) = p ( z ) f ( z ) + p ( z ) f ( z ) + · · · + p n ( z ) f n ( z );return [ p , p , . . . , p n ]; ENDExample 5 ( E singularity) . Let f ( x, y ) = x + y + xy and let F = [ ∂f∂x ( x, y ) , ∂f∂y ( x, y )] . Note that f ( x, y ) is a semi quasi-homogeneous function with respectto the weight vector (7 , . Let ≻ be the weighted degree lexicographical order-ing on K [ ξ, η ] with respect to the weight vector (7 , , where ξ, η correspondto x, y. Then, dim K ( H F ) = 12 , the Milnor number at the origin (0 ,
0) of the curve { ( x, y ) ∈ C | f ( x, y ) = 0 } . The algorithm for computing Grothendieck localduality, mentioned in this section, outputs a basis Ψ F that consists of thefollowing 12 local cohomology classes;1 , η, ξ, η , ξη, η , ξη , η , η − ξ , ξη , ξη − η + ξ η,ξη − η − ξ + ξ η . Note for instance that the local cohomology class xy − x y repre-10ented by ψ (0 , = η − ξ above acts on a holomorphic function h ( x, y ) = P ( i,j ) c ( i,j ) x i y j byres O ( h ( x, y ) , ψ (0 , dx ∧ dy ) = c (0 , − c (2 , . The output implies thatΛ F = { (0 , , (0 , , (0 , , (1 , , (0 . , (1 , , (0 , , (1 , , (0 , , (1 , , (1 , , (1 , } and M F = { x i y j | ( i, j ) ∈ Λ F } is the monomial basis of the quotient space K { x, y } /I F with respect to the local term ordering ≻ − on K { x, y } , where I F denote the ideal in K { x, y } generated by ∂f∂x , ∂f∂y . Furthermore W F = { ψdx ∧ dy | ψ ∈ Ψ F } is the dual basis of the monomial basis M F with respect to theGrothendieck local residue pairing. Since λ F = (3 , x , y ∈ I F . Let J F be the ideal in K [ x, y ] generated by the two polynomials ∂f∂x , ∂f∂y . Let J F,O be the primary component of J F whose associated prime is the maximalideal h x, y i . A Gr¨obner basis of the ideal quotient J F,O : J F is3125 x + 151263 , y + 147 . Set q ( x, y ) = 25 y + 147 . Then, the algorithm localexpression outputs thefollowing: q ( x, y ) x = (49 x + 25 / x y − / y ) ∂f∂x + ( − / xy + 7 / y ) ∂f∂y ,q ( x, y ) y = 25 y ∂f∂x + ( − x + 21 y ) ∂f∂y . Let τ F denote the local cohomology class in H F defined to be τ F = f ( z ) f ( z ) · · · f n ( z ) . Since ω F = τ F dz, the local cohomology class τ F is the kernel function of thepoint residue mapping.Let q ( z ) z m i i = p i, ( z ) f ( z ) + p i, ( z ) f ( z ) + · · · + p i,n ( z ) f n ( z ) , i = 1 , , . . . , n, and set Det( z ) = det( p i,j ( z )) ≤ i,j ≤ n . I M be the ideal in K [ z ] generated by z m , z m , . . . , z m n n . Let u ( z ) ∈ K [ z ]be a polynomial such that u ( z ) q ( z ) − ∈ I M .Since A ( z ) = det( p i,j ( z ) /q ( z )) ≤ i,j ≤ n is equal to q ( z ) n Det( z ) , the transforma-tion law implies the following τ F = u ( z ) n Det( z ) z m z m · · · z m n n . Let λ F = ( ℓ F, , ℓ F, , . . . , ℓ F,n ) . Since m i = ℓ F,i + 1 , the formula above can berewritten as τ F = u ( z ) n Det( z ) ∗ ξ λ F . Note that, according to an algorithm in (Sato and Suzuki, 2009) discoveredby Y. Sato and A. Suzuki, the inverse u ( z ) of q ( z ) in K [ z ] /I M can be obtainedby using Gr¨obner basis computation.The following algorithm computes a representation of the local cohomologyclass τ F , the kernel function of the point residue mapping. Algorithm 2 . tauInput: V = [ z , z , . . . , z n ] , ≻ , F = [ f ( z ) , f ( z ) , . . . , f n ( z )] . /* V : a list of variables, ≻ : a term order */ Output: τ F = P α ∈ Λ F b α ψ α . BEGIN step 1: compute a basis Ψ F = { ψ α | α ∈ Λ F } of the space H F ;/* Λ F : the set of head terms of Ψ F */step 2: compute ℓ F,i = max { ℓ | ξ ℓi ∈ T F } and set m i = ℓ F,i + 1 , i = 1 , , . . . , n ;/* T F = { ξ λ | λ ∈ E F } */step 3: compute a Gr¨obner basis of the ideal J F,O = { p ( z ) ∈ K [ z ] | p ( z ) ∗ ψ α = 0 , α ∈ Λ F } ;step 4: compute G F , R F , S F ;/* notations are from subsection 3.2*/step 5: compute a Gr¨obner basis G Q of the quotient ideal Q = J F : J F,O andchoose a polynomial q ( z ) from G Q such that q ( O ) = 0;step 6: compute q ( z ) z m i i = p i, ( z ) f ( z ) + p i, ( z ) f ( z ) + · · · + p i,n ( z ) f n ( z ) , ( i = 1 , , . . . , n ) , by using the algorithm localexpression ;step 7: compute Det( z ) = det( p i,j ( z )) ≤ i,j ≤ n and set ND = NF I M (Det( z )) , the normal form of Det( z ) with respect to I M ;12tep 8: compute a Gr¨obner basis of the ideal in K [ z, u ] generated by1 − q ( z ) u, z m , z m , . . . , z m n n with respect to an elimination ordering to eliminate u ;step 9: choose a polynomial of degree one with respect to u , of the form cu + poly ( z ), from the Gr¨obner basis of step 8 and setDen = ( − c ) n , NU = NF I M ( poly ( z ) n ) , Num = NF I M (ND × NU);step10: compute ψ = Num ∗ ξ λ F and set Coeff = { c α | α ∈ Λ F } ;/* c α is the coefficient of a term ξ α of ψ , α ∈ Λ F . */return [Λ F , Ψ F , Coeff , Den];
END
The return of the algorithm above means τ F = 1Den X α ∈ Λ F c α ψ α . Note that, since,res O ( h ( z ) τ F dz ) = 1Den X α ∈ Λ F b α | res O ( h ( z ) ψ α dz )holds, the output of the algorithm above completely describes the Grothendieckpoint residue mapping h ( z ) −→ res { O } h ( z ) dzf ( z ) f ( z ) · · · f n ( z ) ! . Let Res F = tau ( V, ≻ , F ) be the output of the algorithm tau . The followingalgorithm residues evaluates the value of Grothendieck point residue. Algorithm 3 . residuesInput: h ∈ K [ z ] , Res F . Output: res { O } ( h ( z ) τ F dz ) . BEGIN step 1: compute the normal form of h by using Ψ F , i.e., NF ≻ ( h )( z ) = X α ∈ Λ F h α z α ;step 2: compute sum = X α ∈ Λ F h α c α ; 13eturn sumDen ; END
Note that NF ≻ ( h ) is computed by the algorithms given in (Tajima and Nakamura,2009; Tajima et al., 2009). The algorithm is free from standard bases compu-tation. All the algorithms given in the present paper are implemented in acomputer algebra system Risa/Asir(Noro and Takeshima, 1992)). Example 6 ( E singularity) . Let us continue the computation. Since step 1to step 6 are done, we start from step 7. From p , p , p , p , = / x y + 49 x − / xy + 7 / y y − x + 21 y , we have the determinantDet = ( − y − x +(175 y +1029 y ) x +(125 / y +245 y ) x − / y − y . A Gr¨obner basis of the ideal in K [ x, y, u ] generated by 1 − uq ( x, y ) , x , y withrespect to a elimination ordering u ≻ x, y is { x , y , − y + 35888671875 y − y + 1240829296875 y − y +42900928441875 y − y − u + 1483273860320763 } . We haveNum = (6654091109227055694580078125 y − y +230061207646859914406689453125 y − y +7954228217665593424662643828125 y − y +275012668088857293301656112771125 y − x +( − y +1893863861348950815395867578125 y − y + 65479206687823165071822883993125 y − y +2263904283707473238459233120331901 y ) x + (15590287306624563112338781903125 y − y +539024829454160294871245981031405 y ) x − y and Den = (218041257467152161) . Since b α = c α Den , we have τ F = 1Den (Num ∗ ( ξ η )) . Therefore, 14 F = 30517578125 / − / η +48828125 / η − / η +78125 / η − / η +125 / η − / η − / ξ +390625 / ξη − / ξη + 625 / ξη − / ξη + 1 / ξη +3125 / ξ − / ξ η + 5 / ξ η − / ξ . This yields τ F = X ≤ i,j ≤ b i,j ψ i,j , where b , = 30517578125 / ,b , = − / ,b , = 48828125 / , b , = − / ,b , = 78125 / , b , = − / ,b , = − / , b , = 390625 / ,b , = − / , b , = 625 / , b , = − / , b , = 1 / . and ψ , = 1 , ψ , = η, ψ , = η , ψ , = η , ψ , = η , ψ , = η − ξ ,ψ , = ξ, ψ , = ξη, ψ , = ξη , ψ , = ξη ,ψ , = ξη − η + 521 ξ η, ψ , = ξη − η − ξ + 521 ξ η . Let NF ≻ ( h )( x, y ) = P ( i,j ) ∈ Λ F h i,j x i y j . Then,res { O } ( h ( x, y ) , τ F dx ∧ dy ) = X ( i,j ) ∈ Λ F h i,j b i,j . We have for instance, res { O } dx ∧ dy ∂f∂x ∂f∂y = 30517578125218041257467152161 . Recall that , as local cohomology class ω F = τ F dx ∧ dy is in H { (0 , } (Ω X ) , thecohomology class τ F defines the residue mappingres { O } ( ∗ , τ F ) : O X,O −→ C . Therefore, the formula above is valid for germs of holomorphic functions h ( x, y ). More precisely, for a germ of holomorphic function h ( x, y ) = X ( i,j ) c i,j x i y j ,we haveres { O } ( h ( x, y ) , τ F dx ∧ dy ) = c , b , + c , b , + c , b , + c , b , + c , b , + c , b , + c , b , + c , b , + ( c , − c , ) b , + c , b , + ( c , − c , + c , ) b , +( c , − c , − c , + c , ) b , . µ -constant deformation In this section, we consider a µ -constant deformation of a quasi homogeneoussingularity, a family of semi-quasi homogeneous isolated hypersurface singu-larities (Greuel, 1986; Lˆe and Ramanujam, 1976). We give, as an applicationof the algorithms presented in the previous section, an algorithm for comput-ing Grothendieck point residues associated to a µ -constant deformation of aquasi homogeneous isolated hypersurface singularity. The keys of the resultingalgorithm are the use of parametric local cohomology systems and parametricGr¨obner systems (comprehensive Gr¨obner systems).Let w = ( w , w , . . . , w n ) ∈ N n be a weight vector for z = ( z , z , . . . , z n ) . Let d w ( z λ ) denote the weighted degree of a monomial z λ = z ℓ z ℓ · · · z ℓ n n definedto be d w ( z λ ) = ℓ w + ℓ w + · · · + ℓ n w n . Definition 7. (1) A non-zero polynomial f is called a weighted homoge-neous (or quasi homogeneous) polynomial of type ( d, w ), if all monomialsof f have the same weighted degree d with respect to the weight vector w , that is f = P d w ( z λ )= d c λ z λ where c λ ∈ K .(2) A polynomial f ( z ) = f ( z ) + g ( z ) is called a semi weighted homogeneous(or semi quasi homogeneous) polynomial of type ( d, w ), if(i) f is weighted homogeneous of type ( d, w ), and f ( z ) = 0 has an isolatedsingularity at the origin O , and(ii) g ( z ) = P d w ( z βj ) >d b j z β j , where b j are coefficients.Let t = ( t , t , . . . , t m ) denote a set of new indeterminates, and let T = { t | t ∈ C m } . Let f t ( z ) = f ( z ) + g ( z, t ) , with g ( z, t ) = X d w ( z βj ) >d t j z β j be a family of semi weighted homogeneous polynomials in K ( t )[ z ], where t ∈ T is regarded as a deformation parameter. Then f t is a µ -constant deformationof f . Set F = [ ∂f∂z , ∂f∂z , . . . , ∂f∂z n ] . Let I F denote a family of ideals in K ( t ) { z } gener-ated by F with the parameter t ∈ T and let H F = ( ψ ∈ H n { O } ( K ( t )[ z ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z ∗ ψ = ∂f∂z ∗ ψ = · · · = ∂f∂z n ∗ ψ = 0 ) . Let ≻ be a term ordering on K ( t )[ ξ ] = K ( t )[ ξ , ξ , . . . , ξ n ] compatible with theweight vector w. It is known, for semi weighted homogeneous cases, that the16et of leading exponents Λ F is independent of t and thus so is the correspond-ing basis monomial set M F . In our previous papers (Nabeshima and Tajima,2015c,b), an algorithm for computing a basis Ψ F of H F is given. The algorithmalso computes Grothendieck local duality as in the non parametric cases. Theother steps, from step 3 to step 10 in the algorithm tau are also executableby using parametric Gr¨obner systems. The step 1 and step 2 of the algorithm residues are also executable.Here we give an example of computation. Example 8 ( E singularity) . Let us consider f = x + y + txy ( t = 0). step 1: A basis Ψ F of the vector space H F with respect to a term ordering ≻ compatible with the weight w = (7 ,
3) is n , η, η , ξ, η , ξη, η , ξη , η − t ξ , ξη , ξη − t η + t ξ η,ξη − t ξ − t η + t ξ η o . The set Λ F isΛ F = { (0 , , (0 , , (0 , , (1 , , (0 . , (1 , , (0 , , (1 , , (0 , , (1 , , (1 , , (1 , } . step 2: x , y ∈ I F . step 5: q ( x, y ) = 147 + 25 t y ∈ J F : J F,O . step 6: q ( x, y ) x q ( x, y ) y = (25 / t y + 49) x − / ty , − / t y x + 7 / t y t y − tx + 21 y ∂f∂x∂f∂y . step 7: Det( x, y ) is( − t y − t ) x +(175 t y +1029 y ) x +(125 / t y +245 t y ) x − / t y − ty . step 8: A Gr¨obner basis of h x , y , − q ( x, y ) u i is { y , x , − t y + 35888671875 t y − t y + 1240829296875 t y − t y +42900928441875 t y − t y − u + 1483273860320763 } . tep 9: We haveDen = (218041257467152161) ,poly ( x, y ) = − t y + 35888671875 t y − t y + 1240829296875 t y − t y +42900928441875 t y − t y + 1483273860320763 , NU = − t y + 372629102116715118896484375 t y − t y + 9202448305874396576267578125 t y − t y + 190901477223974242191903451875 t y − t y +2200101344710858346413248902169 , andNum = (6654091109227055694580078125 t y − t y +230061207646859914406689453125 t y − t y +7954228217665593424662643828125 t y − t y +27501266808885729330165611277112 tt y − t ) x +( − t y +1893863861348950815395867578125 t y − t y +65479206687823165071822883993125 t y − t y +2263904283707473238459233120331901 y ) x +(15590287306624563112338781903125 t y − t y +539024829454160294871245981031405 t y ) x − ty . As an output we thus have τ F = X ≤ i,j ≤ b i,j ψ i,j , where b , = 30517578125 t / ,b , = − t / ,b , = 48828125 t / , b , = − t / ,b , = 78125 t / , b , = − t / ,b , = − t / ,b , = 390625 t / , b , = − t / ,b , = 625 t / , b , = − t / , b , = 1 / . and ψ , = 1 , ψ , = η, ψ , = η , ψ , = η , ψ , = η , ψ , = η − t ξ , , = ξ, ψ , = ξη, ψ , = ξη , ψ , = ξη , ψ , = ξη − t η + 5 t ξ η,ψ , = ξη − t η − t ξ + 5 t ξ η . We have, for instance,res { O } dx ∧ dy ∂f∂x ∂f∂y = 30517578125218041257467152161 t . References
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