An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

Journal of Pure and Applied Algebra

Volumn 139, Issue 1, 1999, Pages 201-233

  Oaku, Toshinori ^a   Takayama, Nobuki ^b  

^a YOKOHAMA CITY UNIVERSITY   (Japan)

^b KOBE UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033147584     PISSN: 00224049     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-4049(99)00012-2     Document Type: Article

References (42)

1. W.W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, American Mathematical Society, Providence, RI, 1994.
2. K. Aomoto, N. Kita, Theory of Hypergeometric Functions, Springer, Tokyo, 1994 (in Japanese).
3. A. Assi, F.J. Castro-Jiménez, J.M. Granger, How to calculate the slopes of a script D-module, Composit. Math. 104 (1996) 107-123.
4. I.N. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl. 6 (1972) 26-40.
5. J.E. Björk, Rings of Differential Operators, North-Holland, Amsterdam, 1979.
6. A. Borel et al., Algebraic D-modules, Academic Press, Boston, 1987.
7. F. Castro, Théorème de division pour les opérateurs diffèrentiels et calcul des multiplicités, These 3ème Cycle, Université Paris VII, 1984.
8. F. Castro, Calculs effectifs pour les idéaux d'opérateurs différentiels, Travaux en Cours, vol. 24, Hermann, Paris, 1987, pp. 1-19.
9. D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms, Springer, New York, 1991.
10. P. Deligne, Équations Différentielles à points singuliers réguliers, Lecture Notes in Math., vol. 163, Springer, Berlin, 1970.
11. A. Dimca, Singularities and Topology of Hypersurfaces, Universitext. Springer, New York, 1992.
12. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, New York, 1995.
13. D. Eisenbud, Computation of cohomology, in: W.V. Vasconcelos et al. (Eds.), Computational Methods in Commutative Algebra and Algebraic Geometry, Springer, New York, 1997, pp. 209-216.
14. A. Galligo, Some algorithmic questions on ideals of differential operators, Lecture Notes in Computer Science, vol. 204, Springer, Berlin, 1985, pp. 413-421.
15. A. Grothendieck, On the de Rham Cohomology of Algebraic Varieties, Publication Mathématique IHES 29 (1966) 95-103.
16. R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.
17. R. Hotta, T. Tanisaki, script D-modules and Algebraic Groups, Springer, Tokyo, 1995 (in Japanese).
18. M. Kashiwara, B-functions and holonomic systems - Rationality of roots of b-functions, Invent. Math. 38 (1976) 33-53.
19. M. Kashiwara, On the holonomic systems of linear differential equations, II, Invent. Math. 49 (1978) 121-135.
20. M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., Kyoto Univ. 20 (2) (1984) 319-365.
21. M. Kashiwara, T. Kawai, On the characteristic variety of a holonomic system with regular singularities, Adv. Math. 34 (1979) 163-184.
22. M. Kashiwara, T. Kawai, On holonomic systems of microdifferential equations, III. Publ. Res. Inst. Math. Sci., Kyoto Univ. 17 (1981) 813-979.
23. Ph. Maisonobe, C. Sabbah (Eds.), D-modules cohérents et holonomes, Les Cours du CIMPA, Travaux en Cours 45, Hermann, Paris, 1993.
24. Ph. Maisonobe, C. Sabbah (Eds.), Images directes et constructibilité, Les Cours du CIMPA, Travaux en Cours 46, Hermann, Paris, 1993.
25. Ph. Maisonobe, script D-modules: An overview towards effectivity, in: Computer Algebra and Differential Equations, London Mathematical Society Lecture Note Series, vol. 193, London Mathematical Society, London, 1994, pp. 21-56.
26. B. Malgrange, Le polynome de Bernstein d'une singularité isolée, Lecture Notes in Mathematics, vol. 459, Springer, Berlin, 1975, pp. 98-119.
27. X-modules cohérents, Travaux en Cours 35, Hermann, Paris, 1989.
28. T. Oaku, Algorithmic methods for Fuchsian systems of linear partial differential equations, J. Math. Soc. Japan 47 (1995) 297-328.
29. T. Oaku, Gröbner bases and systems of linear partial differential equations - An introduction to computational algebraic analysis, Sophia Kokyuroku in Mathematics 38, Department of Mathematics, Sophia University, Tokyo, 1994, (in Japanese).
30. T. Oaku, An algorithm of computing b-functions, Duke Math. J. 87 (1997) 115-132.
31. T. Oaku, Algorithms for the b-function and D-modules associated with a polynomial, J. Pure Appl. Algebra 117 & 118 (1997) 495-518.
32. T. Oaku, Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules, Adv. Appl. Math. 19 (1997) 61-105.
33. T. Oaku, N. Takayama, Algorithms for script D-modules - Restriction, Tensor Product, Localization, and Algebraic Local Cohomology Groups, Preprint, 1998, math.AG/9805006 at http://xxx.lanl.gov.
34. P. Orlik, H. Terao, Arrangements of Hyperplanes, Springer, Berlin, 1992.
35. C. Sabbah, Proximité évanescente. II. Équations fonctionnelles pour plusieurs fonctions analytiques, Compositio Math. 64 (1987) 213-241.
36. J.P. Serre, Faisceaux algébriques cohérents, Ann. Math. 61 (1955) 197-278.
37. R. La Scala, M. Stillman, Strategies for computing minimal free resolutions, J. Symbol. Comput. 26 (1998) 409-431.
38. B. Sturmfels, N. Takayama, Gröbner bases and hypergeometric functions, in: B. Buchberger, F. Winkler (Eds.), Gröbner Bases and Applications, Cambridge University Press, Cambridge, 1998, pp. 246-258.
39. N. Takayama, Gröbner basis and the problem of contiguous relations, Jpn. J. Appl. Math. 6 (1989) 147-160.
40. N. Takayama, An algorithm of constructing the integral of a module - An infinite dimensional analog of Gröbner basis, in: S. Watanabe, M. Nagata (Eds.), Proc. of Internat. Symp. on Symbolic and Algebraic Computation, ACM, New York, 1990, pp. 206-211.
41. N. Takayama, Kan: A System for Computation in Algebraic Analysis, 1991 -, Source code available for Unix computers from ftp.math.kobe-u.ac.jp via anonymous ftp. See also www.math.kobe-u.ac.jp/KAN/.
42. U. Walther, Algorithmic computation of local cohomology modules and the cohomological dimension of algebraic varieties, J. Pure Appl. Algebra 139 (1999) 303-321.