Papers in Computational Methods for D-modules
Nobuki Takayama, takayama@math.kobe-u.ac.jp
This is a collection of papers in computational D-module theory. The scopeis limited to computation of invariants of D-modules including functors of homology algebra in D-modules. If there are papers which are not cited, please
let me know them.

References[1] Assi, A., Castro-Jiménez, F.J., Granger, M.: How to calculate the slopesof a D-module. Compositio Mathematica 104, (1996), 107–123.
[2] A. Assi, F.J. Castro-Jiménez, M. Granger: The Gröbner fan of an An module, Journal of Pure and Applied Algebra 150 (2000), 27–39.
[3] A. Assi, F.J. Castro-Jiménez, M. Granger: The analytic standard fan of aD-module, Journal of Pure and Applied Algebra 164 (2001), 3–21.
[4] R. Bahloul: Démonstration constructive de l’existence de polynômes deBernstein-Sato pour plusieurs fonctions analytiques, Compositio Math. 141(2005), 175–191.
[5] Rouchdi Bahloul, Nobuki Takayama: Local Gröbner fan: polyhedral andcomputational approach math.AG/0412044 (2004).
[6] Castro, F. : Théorème de division pour les opérateurs diffèrentiels et calcul
des multiplicités. Thèse de 3ème cycle, Université de Paris 7, 1984.
[7] Castro, F.: Calculs effectifs pour les idéaux d’opérateurs différentiels.
Travaux en Cours, 24, 1–19, Hermann, Paris.
[8] Francisco Castro-Jimenez and Nobuki Takayama: Singularities of the Hypergeometric System associated with a Monomial Curve. Transaction ofthe American Mathematica Society, 355 (2003), 3761–3775.
[9] F.J. Castro-Jiménez, M. Granger: Explicit calculations in rings of differential operators. Éléments de la théorie des systèmes différentielsgéométriques, 89–128, Sémin. Congr., 8, Soc. Math. France, Paris, 2004[10] Edited by A.Cohen, X.Gao and N.Takayama, Mathematical Software Proceedings of ICMS 2002, (2002), World Scientific. There are papers for thespecial session of computational D-module theory.
1[11] Galligo, A. (1985): Some algorithmic questions on ideals of differentialoperators. EUROCAL ’85, Springer Lecture Notes in Computer Science
204, 413–421.
[12] Granger, M. , Oaku,T. : Minimal filtered free resolutions for analytic Dmodules. J. Pure Appl. Algebra 191 (2004), no. 1-2, 157–180.
[13] Leykin, A.: Constructibility of the set of polynomials with a fixedBernstein-Sato polynomial: an algorithmic approach. Effective methods
in rings of differential operators. J. Symbolic Comput. 32 (2001), no. 6,663–675.
[14] Leykin, A.: D-modules for Macaulay 2. Mathematical software (Beijing,2002), 169–179, World Sci. Publishing, River Edge, NJ, 2002.
[15] Leykin, A.: Computing local cohomology in Macaulay 2. Local cohomology
and its applications (Guanajuato, 1999), 195–205, Lecture Notes in Pureand Appl. Math., 226, Dekker, New York, 2002.
[16] Leykin, A.:
Algorithmic
math.RA/0204303

proofsoftwotheorems

ofStafford,[17] Oaku, T.: Computation of the characteristic variety and the singular locus
of a system of differential equations with polynomial coefficients. Japan
Journal of Industrial and Applied Mathematics 11, (1994) 485–497.
[18] Oaku, T. : Algorithms for b-functions, restrictions, and algebraic local
cohomology groups of D-modules. Advances in Applied Mathematics 19,(1997) 61–105.
[19] Oaku, T., Takayama, N.: An algorithm for de Rham cohomology groupsof the complement of an affine variety via D-module computation. Journalof Pure and Applied Algebra, 139, (1999), 201–233.
[20] Oaku, T., Takayama, N. Algorithms for D-modules — Restrictions, tensor
product, localization and algebraic local cohomology groups. Journal ofPure and Applied Algebra 156, (2001) 267–308. math.AG/9805006.
[21] Oaku, T., Takayama, N., Walther, U.: A localization algorithmfor D-modules. Journal of Symbolic Computation 29, (2000) 721–728.
math.AG/9811039.
[22] Oaku, T., Takayama, N. : Minimal Free Resolutions of Homogenized Dmodules. Journal of Symbolic Computation 32 (2001), 575–595.
[23] Oaku, T., Takayama, N., Tsai, H.: Polynomial and Rational Solutions ofHolonomic Systems. Journal of Pure and Applied Algebra 164 (2001), no.
1-2, 199–220.

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[24] Toshinori Oaku, Yoshinao Shiraki, Nobuki Takayama: Algebraic Algorithms for D-modules and Numerical Analysis, Z.M.Li, W.Sit (editors),
Computer Mathematics, Proceedings of the sixth asian symposium, 23–39,
World scientific, 2003.
[25] Michel Granger, Toshinori Oaku, Nobuki Takayama: Tangent cone algorithm for homogenenized differential operators, Preprint.
[26] M. Saito, B. Sturmfels, N. Takayama: Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics
6, Springer-Verlag, Berlin, 2000.
[27] Nobuki Takayama: An Algorithm for Constructing Cohomological Series
Solutions of Holonomic Systems, Journal of Japan Society for Symbolic andAlgebraic Computation, 10, No.4, (2003) 2–11. math.AG/0309378[28] Tsai, Harrison: Algorithms for associated primes, Weyl closure, and local
cohomology of D-modules. Local cohomology and its applications (Guanajuato, 1999), 169–194, Lecture Notes in Pure and Appl. Math., 226, Dekker,New York, 2002.
[29] Tsai, H., Walther, U. : Computing homomorphisms between holonomic Dmodules. Effective methods in rings of differential operators. J. Symbolic
Comput. 32 (2001), no. 6, 597–617.
[30] Tsai, Harrison: Weyl closure of a linear differential operator. Symbolic
computation in algebra, analysis, and geometry (Berkeley, CA, 1998). J.
Symbolic Comput. 29 (2000), no. 4-5, 747–775.
[31] Reid, G. : Algorithms for reducing a system of PDEs to standard form,
determining the dimension of its solutions space and calculating Taylor
series solutions. European Journal of Applied Mathematics 2, (1991) 293–318.
[32] Walther, U. : Algorithmic computation of local cohomology modules andthe cohomological dimension of algebraic varieties. Journal of Pure andApplied Algebra 139 (1999), 303-321.
[33] Walther, U.: Algorithmic computation of de Rham cohomology of complements of complex affine varieties, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). J. Symbolic Comput. 29 (2000),
no. 4-5, 795–839. math.AG/9807176.
[34] Walther, Uli: Computing the cup product structure for complements ofcomplex affine varieties. Effective methods in algebraic geometry (Bath,2000). J. Pure Appl. Algebra 164 (2001), no. 1-2, 247–273.
[35] Walther, Uli: Algorithmic determination of the rational cohomology ofcomplex varieties via differential forms. Symbolic computation: solving
equations in algebra, geometry, and engineering (South Hadley, MA, 2000),
185–206, Contemp. Math., 286, Amer. Math. Soc., Providence, RI, 2001.
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[36] Walther, Uli : D-modules and cohomology of varieties. Computations inalgebraic geometry with Macaulay 2, 281–323, Algorithms Comput. Math.,
8, Springer, Berlin, 2002. 14-04[37] Walther, Uli: Cohomology, stratifications and parametric Gröner bases incharacteristic zero. Computer algebra and computer analysis (Berlin, 2001).
J. Symbolic Comput. 35 (2003), no. 5, 527–542.

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