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An algorithm for computing Grothendieck local residues. II: General case. (English) Zbl 1457.32006
Summary: 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.
For Part I, see [the authors, ibid. 13, No. 1–2, 205–216 (2019; Zbl 07095839)].
32A27 Residues for several complex variables
13N10 Commutative rings of differential operators and their modules
32C38 Sheaves of differential operators and their modules, \(D\)-modules
Full Text: DOI
[1] Ehrenpreis, L., Fourier Analysis in Several Complex Variables (1970), Hoboken: Wiley, Hoboken · Zbl 0195.10401
[2] Gianni, P.; Trager, B.; Zacharias, G., Gröbner bases and primary decomposition of polynomial ideals, J. Symb. Comput., 6, 149-167 (1988) · Zbl 0667.13008
[3] Hartshorne, R., Residues and Duality (1966), Berlin: Springer, Berlin
[4] Hörmander, L., An Introduction to Complex Analysis in Several Variables (1990), Amsterdam: North-Holland, Amsterdam
[5] Kashiwara, M., On the maximally overdetermined system of linear differential equations. I, Publ. Res. Inst. Math. Sci., 10, 563-579 (1975) · Zbl 0313.58019
[6] Kashiwara, M., On the holonomic systems of linear differential equations. II, Invent. Math., 49, 121-135 (1978) · Zbl 0401.32005
[7] Kashiwara, M., On holonomic systems of micro-differential equations. III—Systems with regular singularities, Publ. Res. Inst. Math. Sci., 17, 813-979 (1981) · Zbl 0505.58033
[8] 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) · Zbl 1229.13003
[9] Noro, M. et al.: Risa/Asir a computer algebra system, 1994-2019. http://www.math.kobe-u.ac.jp/Asir/ · Zbl 1027.68152
[10] Oaku, T., Algorithms for the \(b\)-functions, restrictions, and algebraic local cohomology groups of \(D\)-modules, Adv. Appl. Math., 19, 61-105 (1997) · Zbl 0938.32005
[11] Oaku, T.; Takayama, N., Algorithms for \(D\)-modules—restriction, tensor product, localization, and local cohomology groups, J. Pure Appl. Algebra, 156, 267-308 (2001) · Zbl 0983.13008
[12] Ohara, K.; Tajima, S., An algorithm for computing Grothendieck local residues I: shape basis case, Math. Comput. Sci., 13, 205-216 (2019) · Zbl 07095839
[13] Palamodev, VP, Linear Differential Operators with Constant Coefficients (1970), Berlin: Springer, Berlin
[14] 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) · Zbl 1140.32302
[15] Tajima, S., Noether differential operators and Grothendieck local residues, RIMS Kôkyûroku, 1431, 123-136 (2005)
[16] Tajima, S.; Son, LH; Tutschke, W.; Jain, S., An algorithm for computing exponential polynomial solutions of constant coefficients holonomic PDE’s—generic case—, Methods of Complex and Clifford Analysis, 335-344 (2006), Delhi: SAS International Publ, Delhi · Zbl 1107.35034
[17] Tajima, S.; Oaku, T.; Nakamura, Y., Multidimensional local residues and holonomic \(D\)-modules, RIMS Kôkyûroku, 1033, 59-70 (1998) · Zbl 0944.32008
[18] Tajima, S.; Nakamura, Y., Computational aspects of Grothendieck local residues, Séminaires et Congrès, 10, 287-305 (2005) · Zbl 1089.32002
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