Grothendieck local residue is considered in the context of symbolic computation. An effective method based on the theory of holonomic D-modules is proposed for computing Grothendieck local residues. The key is the notion of Noether operator associated to a local cohomology class. The resulting algorithm and an implementation are described with illustrations.
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Ehrenpreis, L.: Fourier Analysis in Several Complex Variables. Wiley, Hoboken (1970)
Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. J. Symb. Comput. 6, 149–167 (1988)
Hartshorne, R.: Residues and Duality. Lecture Notes in Mathematics, vol. 20. Springer, Berlin (1966)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3 revised edn. North-Holland, Amsterdam (1990)
Kashiwara, M.: On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. 10, 563–579 (1975)
Kashiwara, M.: On the holonomic systems of linear differential equations. II. Invent. Math. 49, 121–135 (1978)
Kashiwara, M.: On holonomic systems of micro-differential equations. III—Systems with regular singularities. Publ. Res. Inst. Math. Sci. 17, 813–979 (1981)
Noro, M.: New algorithms for computing primary decomposition of polynomial ideals. In: Mathematical Software—ICMS 2010. Lecture Notes in Computer Science 6327, pp. 233–244. Springer, Berlin (2010)
Noro, M. et al.: Risa/Asir a computer algebra system, 1994–2019. http://www.math.kobe-u.ac.jp/Asir/
Oaku, T.: Algorithms for the \(b\)-functions, restrictions, and algebraic local cohomology groups of \(D\)-modules. Adv. Appl. Math. 19, 61–105 (1997)
Oaku, T., Takayama, N.: Algorithms for \(D\)-modules—restriction, tensor product, localization, and local cohomology groups. J. Pure Appl. Algebra 156, 267–308 (2001)
Ohara, K., Tajima, S.: An algorithm for computing Grothendieck local residues I: shape basis case. Math. Comput. Sci. 13, 205–216 (2019)
Palamodev, V.P.: Linear Differential Operators with Constant Coefficients. Springer, Berlin (1970)
Tajima, S.: On Noether differential operators attached to a zero-dimensional primary ideal—a shape basis case—. In: Finite or Infinite Dimensional Complex Analysis and Applications, pp. 357–366. Kyushu Univ. Press, Fukuoka (2005)
Tajima, S.: Noether differential operators and Grothendieck local residues. RIMS Kôkyûroku 1431, 123–136 (2005). (in Japanese)
Tajima, S.: An algorithm for computing exponential polynomial solutions of constant coefficients holonomic PDE’s—generic case—. In: Son, L.H., Tutschke, W., Jain, S. (eds.) Methods of Complex and Clifford Analysis, pp. 335–344. SAS International Publ, Delhi (2006)
Tajima, S., Oaku, T., Nakamura, Y.: Multidimensional local residues and holonomic \(D\)-modules. RIMS Kôkyûroku 1033, 59–70 (1998). (in Japanese)
Tajima, S., Nakamura, Y.: Computational aspects of Grothendieck local residues. Séminaires et Congrès 10, 287–305 (2005)
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This work has been partly supported by JSPS KAKENHI Grant Numbers JP17K05292, JP18K03320 and by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
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Ohara, K., Tajima, S. An Algorithm for Computing Grothendieck Local Residues II: General Case. Math.Comput.Sci. (2020). https://doi.org/10.1007/s11786-019-00439-y
- Local residues
- Local cohomology
- Holonomic D-modules
- Noether operators
Mathematics Subject Classification
- Primary 32A27
- Secondary 13N10