Proceedings of the Second ISAAC Congress pp 809-817 | Cite as

# An Algorithm for the Local Residues with the Viewpoint of D-Module

## Abstract

The local residue has been presented since the early days of several complex variables and plays a fundamental role in complex analysis and geometry. Nevertheless, in many cases, there are very few practical tools in the computational point of view.

The purpose of this note is to give a new algorithm for computing the local residue with the viewpoint of the theory of *D-*modules. For a given regular sequence of holomorphic functions, consider the algebraic local cohomology group with support at the set of common zeros of these functions. If the set of common zeros consists of finitely many points, the cohomology classes with support at each zero can be characterized as a solution of a system of certain linear partial differential equations. In this note, a system of such linear partial differential equations is constructed by using the transformation formula. And, on the basis of the properties of the image of adjoint operators of the system, an algorithm for computing local residues is provided.

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### References

- [1]M. Kashiwara,
*On the holonomic systems of linear differntial eqauations*,*II*. Inventiones mathematicae**49**(1978), pp. 121–135.MathSciNetMATHCrossRefGoogle Scholar - [2]T. Oaku,
*Algorithms for the b-functions*,*restrictions*,*and algebraic local cohomology groups of D-modules*, Adv. in Appl. Math.**19**(1997), pp. 61–105.MathSciNetMATHCrossRefGoogle Scholar - [3]S. Tajima and Y. Nakamura,
*Computational aspects of local residues of a function of several variables*, preprint.Google Scholar - [4]S. Tajima, Y. Nakamura,
*Residue calculus with Differential operator*, Kyushu J. Math.**53**(1999), to appear.Google Scholar - [5]S. Tajima and Y. Nakamura,
*About computatoin of residue of several variables function*(in Japanese), J. JSSAC,**7**, 1 (1998), 39–40.Google Scholar - [6]S. Tajima and Y. Nakamura,
*Residue calculus with D module (the case of shape lemma)*(in Japanese), J. JSSAC, to appear.Google Scholar - [7]S. Tajima and Y. Nakamura,
*Localization of residue calculus with D module(in*Japanese), J. JSSAC,**7**, 3 (1999), to appear.Google Scholar - [8]S. Tajima, T. Oaku and Y. Nakamura,
*Multidi-**mensional local residues and holonomic D-modules*, Sûrikaiseki Kenkyûshokôyûroku,**1033**(1998), pp. 59–70.MathSciNetMATHGoogle Scholar - [9]N. Takayama,
*Kan: A system for computation in algebraic analysis*(1991-),.http://www.math.s.kobe-u.ac.jp - [10]K. Vilonen,
*Intersection homology D-module on local complete intersections with isolated singularities*, Invent. math.**81**, (1985), pp. 107–114.MathSciNetMATHCrossRefGoogle Scholar