Some algorithmic aspects of the D-module theory

  • Toshinori Oaku


We consider D-modules from an algorithmic point of view. Our aim is to present algorithms for computing some invariants attached to a D-module, such as the characteristic variety, the multiplicity, the b-function (or exponents), and the induced system. In particular, we obtain algorithms for computing the classical b-function (Bernstein-Sato polynomial) and D-modules associated with an arbitrary polynomial.


Left Ideal Computer Algebra System Principal Symbol Monic Polynomial Free Resolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BW]
    Becker, T., Weispfenning, V., Gröbner Bases, Springer-Verlag, Berlin, 1993.MATHCrossRefGoogle Scholar
  2. [Be]
    Bernstein, I. N., Modules over the ring of differential operators, Functional Anal. Appl. 2 (1971), 1–16.CrossRefGoogle Scholar
  3. [Bj]
    Björk, J.E., Rings of Differential Operators, North-Holland, Amsterdam, 1979.MATHGoogle Scholar
  4. [Bo]
    Borel, A. et al., Algebraic D-Modules, Academic Press, Boston, 1987.MATHGoogle Scholar
  5. [BGMM]
    Briançon, J., Granger, M., Maisonobe, Ph., Miniconi, M., Algorithme de calcul du polynôme de Bernstein: cas non dégénéré, Ann. Inst. Fourier 39 (1989), 553–610.MATHCrossRefGoogle Scholar
  6. [Bu]
    Buchberger, B., Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), 374–383.MathSciNetMATHCrossRefGoogle Scholar
  7. [C]
    Castro, F., Calculs effectifs pour les idéaux d’opérateurs différentiels, Travaux en Cours 24, Hermann, Paris, 1987, pp. 1–19.Google Scholar
  8. [CLO]
    Cox, D., Little, J., O’Shea, D., Ideals, Varieties, and Algorithms, Springer, Berlin, 1992.MATHGoogle Scholar
  9. [E]
    Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Springer, New York, 1995.MATHGoogle Scholar
  10. [G]
    Galligo, A., Some algorithmic questions on ideals of differential operators, Lecture Notes in Comput. Sci. 204, Springer, Berlin, 1985, pp. 413–421.Google Scholar
  11. [K1]
    Kashiwara, M., Algebraic study of systems of linear partial differential equations (in Japanese), Master’s thesis, University of Tokyo, 1971.Google Scholar
  12. [K2]
    Kashiwara, M., B-functions and holonomic systems-Rationality of roots of b-functions, Invent. Math. 38, 1976, 33–53.MathSciNetMATHCrossRefGoogle Scholar
  13. [K3]
    Kashiwara, M., On the holonomic systems of linear differential equations, II, Invent. Math. 49 (1978), 121–135.MathSciNetMATHCrossRefGoogle Scholar
  14. [K4]
    Kashiwara, M., Systems of Microdifferential Equations, Progress in Math. 34, Birkhäuser, Boston, 1983.Google Scholar
  15. [K5]
    Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations, Lecture Notes in Math. 1016, Springer, Berlin, 1983, pp. 134–142.Google Scholar
  16. [KK]
    Kashiwara, M., Kawai, T., Second microlocalization and asymptotic expansions, Lecture Notes in Physics 126, Springer, Berlin, 1980, pp. 21–76.Google Scholar
  17. [L1]
    Laurent, Y., Calcul d’indices et irrégularité pour les systèmes holonomes, Astérisque 130 (1985), 352–364.MathSciNetGoogle Scholar
  18. [L2]
    Laurent, Y., Polygône de Newton et b-fonctions pour les modules microdifferentiels, Ann. Sci. Éc. Norm. Sup. 20 (1987), 391–441.MathSciNetMATHGoogle Scholar
  19. [LS]
    Laurent, Y., Schapira, P., Images inverses des modules différentiels, Compositio Math. 61 (1987), 229–251.MathSciNetMATHGoogle Scholar
  20. [La]
    Lazard, D., Gröbner bases, Gaussian elimination, and resolution of systems of algebraic equations, Lecture Notes in Comput. Sci. 162, Springer, Berlin, 1983, pp. 146–156.Google Scholar
  21. [M1]
    Malgrange, B., Le polyôme de Bernstein d’une singularité isolée, Lecture Notes in Math. 459, Springer, Berlin, 1975, pp. 98–119.Google Scholar
  22. [M2]
    Malgrange, B., Polynômes de Bernstein-Sato et cohomologie évanescente, Astérisque 101–102 (1983), 243-267.Google Scholar
  23. [Mo]
    Mora, F., An algorithm to compute the equations of tangent cones, Lecture Notes in Comput. Sci. 144, Springer, Berlin, 1982, pp. 158–165.Google Scholar
  24. [N]
    Noumi, M., Wronskian determinants and the Gröbner representation of a linear differential equation, in Algebraic Analysis (M. Kashiwara, T. Kawai, eds.), Academic Press, Boston, 1988, pp. 549–569.Google Scholar
  25. [NT]
    Noro, M, Takeshima T., Risa/Asir-a computer algebra system, in Proceedings of International Symposium on Symbolic and Algebraic Computation (Paul S. Wnag, ed.), ACM, New York, 1992, pp. 387–396.Google Scholar
  26. [O1]
    Oaku, T., Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Japan J. Indust. Appl. Math. 11 (1994), 485–497.MathSciNetMATHCrossRefGoogle Scholar
  27. [O2]
    Oaku, T., Algorithms for finding the structure of solutions of a system of linear partial differential equations, in Proceedings of International Symposium on Symbolic and Algebraic Computation (J. Gathen, M. Giesbrecht, eds.), ACM, New York, 1994, pp. 216–223.CrossRefGoogle Scholar
  28. [O3]
    Oaku, T., Gröbner Bases and Systems of Linear Partial Differential Equations, Sophia Kokyuroku in Mathematics, No. 38 (in Japanese), Department of Mathematics, Sophia University, Tokyo, 1994.Google Scholar
  29. [O4]
    Oaku, T., Algorithmic methods for Fuchsian systems of linear partial differential equations, J. Math. Soc. Japan 47 (1995), 297–328.MathSciNetMATHCrossRefGoogle Scholar
  30. [O5]
    Oaku, T., Gröbner bases for D-modules on a non-singular affine algebraic variety, To appear in Tôhoku Math. J.Google Scholar
  31. [O6]
    Oaku, T., An algorithm of computing b-functions, (preprint).Google Scholar
  32. [O7]
    Oaku, T., Computation of b-functions and induced systems of D-modules, (in preparation).Google Scholar
  33. [O8]
    Oaku, T., Algorithms for the b-function and D-modules associated with an arbitrary polynomial, (in preparation).Google Scholar
  34. [SKK]
    Sato, M., Kawai, T., Kashiwara, M., Microfunctions and pseudo-differential equations, Lecture Notes in Math. 287, Springer, Berlin, 1973, pp. 265–529.Google Scholar
  35. [SKKO]
    Sato, M., Kashiwara, M., Kimura, T., Oshima, T., Micro-local analysis of prehomogeneous vector spaces, Invent. Math. 62 (1980), 117–179.MathSciNetMATHCrossRefGoogle Scholar
  36. [S]
    Schapira, P., Microdifferential Systems in the Complex Domain, Grundlehren der Math. Wiss. vol. 269, Springer, Berlin, 1985.Google Scholar
  37. [T1]
    Takayama, N., Gröbner basis and the problem of contiguous relations, Japan J. Appl. Math. 6 (1989), 147–160.MathSciNetCrossRefGoogle Scholar
  38. [T2]
    Takayama, N., An algorithm of constructing the integral of a module —an infinite dimensional analog of Gröbner basis, in Proceedings of International Symposium on Symbolic and Algebraic Computation (S. Watanabe, M. Nagata, eds.), ACM, New York, 1990, pp. 206–211.CrossRefGoogle Scholar
  39. [T3]
    Takayama, N., An approach to the zero recognition problem by Buchberger algorithm, J. Symbolic Comput. 14 (1992), 265–282.MathSciNetMATHCrossRefGoogle Scholar
  40. [T4]
    Takayama, N., Kan: A system for computation in algebraic analysis 1991.Google Scholar
  41. [Y]
    Yano, T., On the theory of b-functions, Publ. RIMS, Kyoto Univ. 14 (1978), 111–202.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1997

Authors and Affiliations

  • Toshinori Oaku
    • 1
  1. 1.Department of Mathematical SciencesYokohama City UniversityYokohama, Kanagawa 236Japan

Personalised recommendations