Advertisement

Mathematics in Computer Science

, Volume 13, Issue 1–2, pp 205–216 | Cite as

An Algorithm for Computing Grothendieck Local Residues I: Shape Basis Case

  • Katsuyoshi OharaEmail author
  • Shinichi Tajima
Article
  • 33 Downloads

Abstract

Grothendieck local residue is considered in the context of symbolic computation. Basic ideas of our approach are the use of local cohomology, holonomic D-modules, and Noether operators. An effective method is introduced for computing Grothendieck local residues of rational n-forms under shape basis condition. Resulting algorithms that avoid the use of Gröbner bases on the Weyl algebra and an implementation are described. Some examples are also given for illustration.

Keywords

Local residues Local cohomology Holonomic system Computer algebra 

Mathematics Subject Classification

Primary 32A27 Secondary 13N10 

Notes

References

  1. 1.
    Altman, A., Kleiman, S.: Introduction to Grothendieck Duality Theory. Lecture Notes in Mathematics, vol. 146. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ehrenpreis, L.: Fourier Analysis in Several Complex Variables. Wiley, New York (1970)zbMATHGoogle Scholar
  3. 3.
    Gianni, P., Mora, T.: Algebraic solution of systems of polynomial equation using Gröbner bases. In: Proceedings of the AAECC 5, LNCS vol. 356, pp. 247–257 (1989)Google Scholar
  4. 4.
    Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. J. Symb. Comput. 6, 149–167 (1988)CrossRefzbMATHGoogle Scholar
  5. 5.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  6. 6.
    Hartshorne, R.: Residues and Duality. Lecture Notes in Mathematics, vol. 20. Springer, Berlin (1966)CrossRefGoogle Scholar
  7. 7.
    Kashiwara, M.: On the maximally overdetermined system of linear differential equations. I. Res. Inst. Math. Sci. 10, 563–579 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kashiwara, M.: On the holonomic systems of linear differential equations. II. Invent. Math. 49, 121–135 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kashiwara, M.: On holonomic systems of micro-differential equations. III: systems with regular singularities. Res. Inst. Math. Sci. 17, 813–979 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nakamura, Y., Tajima, S.: Residue calculus with differential operator. Kyushu J. Math. 54, 127–138 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Noro, M.: New algorithms for computing primary decomposition of polynomial ideals. In: Mathematical Software—ICMS 2010. Lecture Notes in Computer Science, vol. 6327, pp. 233–244. Springer, Berlin (2010)Google Scholar
  12. 12.
    Noro, M., et al.: Risa/Asir a computer algebra system, 1994–2018. http://www.math.kobe-u.ac.jp/Asir/
  13. 13.
    Oaku, T.: Algorithms for the \(b\)-functions, restrictions, and algebraic local cohomology groups of \(D\)-modules. Adv. Appl. Math. 19, 61–105 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Oaku, T., Takayama, N.: Algorithms for \(D\)-modules—restriction, tensor product, localization, and local cohomology groups. J. Pure Appl. Algebra 156, 267–308 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    OpenXM committers: OpenXM, a project to integrate mathematical software systems. 1998–2018. http://www.openxm.org
  16. 16.
    Pham, F.: Singularités des Systèms Différentielles de Gauss–Manin. Birkhäuser, Basel (1979)Google Scholar
  17. 17.
    Palamodev, V.P.: Linear Differential Operators with Constant Coefficients. Springer, New York (1970)CrossRefGoogle Scholar
  18. 18.
    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 University Press, Fukuoka (2005)Google Scholar
  19. 19.
    Tajima, S.: Noether differential operators and Grothendieck local residues. RIMS Kôkyûroku 1432, 123–136 (2005) (in Japanese) Google Scholar
  20. 20.
    Tajima, S., Oaku, T., Nakamura, Y.: Multidimensional local residues and holonomic \(D\)-modules. RIMS Kôkyûroku 1033, 59–70 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tajima, S., Nakamura, Y.: Computational aspects of Grothendieck local residues. Séminaires et Congrès 10, 287–305 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsKanazawa UniversityKanazawaJapan
  2. 2.Graduate School of Science and TechnologyNiigata UniversityNiigataJapan

Personalised recommendations