zbMATH — the first resource for mathematics

Algorithms for \(b\)-functions, restrictions, and algebraic local cohomology groups of \(D\)-modules. (English) Zbl 0938.32005
From the introduction: “Our purpose is to present algorithms for computing some invariants and functors attached to algebraic \(D\)-modules by using Gröbner bases for differential operators. Let \(K\) be an algebraically closed field of characteristic zero and let \(X\) be a Zariski open set of \(K^n\) with a positive integer \(n\). We fix a coordinate system \(x=(x_1, \dots, x_n)\) of \(X\) and write \(\partial= (\partial_1, \dots, \partial_n)\) with \(\partial_i:= \partial/ \partial x_i\). We denote by \({\mathcal D}_X\) the sheaf of algebraic linear differential operators on \(X\).
Let \({\mathcal M}\) be a coherent left \({\mathcal D}_X\)-module and \(u\) a section of \({\mathcal M}\). Suppose that \(f=f(x) \in K[x]\) is an arbitrary nonconstant polynomial of \(n\) variables. If \({\mathcal M}\) is holonomic, then for each point \(p\) of \(Y:=\{x\in X|f(x)=0\}\), there exists a germ \(P(x,\partial,s)\) of \({\mathcal D}_X[s]\) at \(p\) and a polynomial \(b(s) \in K[s]\) of one variable so that \[ P(x,\partial,s)(f^{s+1}u)= b(s)f^su\tag{1.1} \] holds with an indeterminate \(s\). More precisely, (1.1) means that there exists a nonnegative integer \(m\) so that \[ Q:=f^{m-s}\bigl(b(s)-P(s,\partial, s) f \bigr)f^s\in {\mathcal D}_X[s] \] satisfies \(Qu=0\) in \({\mathcal M}[s]:=K[s] \otimes_K {\mathcal M}\). The monic polynomial \(b(s)\) of the least degree that satisfies (1.1), if any, is called the (generalized) \(b\)-function for \(f\) and \(u\) at \(p\). Suppose that a presentation (i.e., generators and the relations among them) of a coherent left \({\mathcal D}_X\)-module \({\mathcal M}\) and a section \(u\) of \({\mathcal M}\) are given. Then we are concerned with algorithms for solving the following problems:
(A1) to determine whether there exists and to find, if it does, the \(b\)-function for \(f\) and \(u\);
(A2) to obtain presentations of the algebraic local cohomology groups \({\mathcal H}^j_{[Y]} ({\mathcal M})\) \((j=0,1)\) as left \({\mathcal D}_X\)-modules;
(A3) to obtain a presentation of the localization \({\mathcal M}(*Y)= {\mathcal M}[f^{-1}]\) of \({\mathcal M}\) by \(f\) as a left \({\mathcal D}_X\)-module;
(A4) to obtain a presentation of the left \({\mathcal D}_X[s]\)-module \(\sum_{i= 1}^r {\mathcal D}_X[s] (f^s\otimes u_i)\), where \(u_1,\dots,u_r\) are generators of \({\mathcal M}\) and \(f^s\otimes u_i\) is regarded as a section of \(({\mathcal O}_X [s,f^{-1}]f^s) \otimes_{{\mathcal O}_X}{\mathcal M}\)”.

32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
PDF BibTeX Cite
Full Text: DOI
[1] Assi, A.; Castro-Jiménez, F-J.; Granger, J-M., Détermination des pentes de \(D\)-modules, C.R. acad. sci. Paris, 320, 193-198, (1995) · Zbl 0849.13019
[2] Becker, T.; Weispfenning, V., Gröbner bases, (1993), Springer-Verlag Berlin
[3] Björk, J.E., Rings of differential operators, (1979), North-Holland Amsterdam
[4] Briançon, J.; Granger, M.; Maisonobe, Ph.; Miniconi, M., Algorithme de calcul du polynôme de Berstein: cas non dégénéré, Ann. inst. Fourier, 39, 553-610, (1989) · Zbl 0675.32008
[5] Briançon, J.; Maisonobe, Ph.; Merle, M., Localisation de systèms différentiels, stratifications de Whitney et condition de thom, Invent. math., 117, 531-550, (1994) · Zbl 0920.32010
[6] Borel, A., AlgebraicD, (1987), Academic Press Boston · Zbl 0642.32001
[7] Buchberger, B., Ein algorithmisches kriterium für die Lösbarkeit eines algebraischen gleichungssystems, Aequationes math., 4, 374-383, (1970) · Zbl 0212.06401
[8] Castro, F., Calculs effectifs pour LES idéaux d’operateurs différentiels, Travaux en cours, (1987), Hermann Paris, p. 1-19 · Zbl 0633.13009
[9] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms, (1992), Springer-Verlag Berlin
[10] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (1995), Springer-Verlag New York · Zbl 0819.13001
[11] Eisenbud, D.; Huneke, C.; Vasconcelos, W., Direct methods for primary decomposition, Invent. math., 110, 207-235, (1992) · Zbl 0770.13018
[12] Gailligo, A., Some algorithmic questions on ideals of differential operators, Lecture notes in computer science, 204, (1985), Springer-Verlag Berlin, p. 413-421
[13] Gelfand, I.M.; Zelevinsky, A.V.; Kapranov, M.M., Hypergeometric functions and toric manifolds, Funct. anal. appl., 23, 94-106, (1989) · Zbl 0721.33006
[14] Ginsburg, V., Characteristic varieties and vanishing cycles, Invent math., 84, 327-402, (1986) · Zbl 0598.32013
[15] Hartshorne, R., Residues and duality, Lecture notes in mathematics, 20, (1966), Springer-Verlag Berlin
[16] Kashiwara, M., Bb, Invent. math., 38, 33-53, (1976)
[17] Kashiwara, M., On the holonomic systems of linear differential equations. II, Invent. math., 49, 121-135, (1978) · Zbl 0401.32005
[18] Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations, Lecture notes in math., 1016, (1983), Springer-Verlag Berlin, p. 134-142
[19] Kashiwara, M., Systems of microdifferential equations, (1983), Birkhäuser Boston · Zbl 0566.32022
[20] Kashiwara, M.; Kawai, T., On the characteristic variety of a holonomic system with regular singularities, Adv. in math., 34, 163-184, (1979) · Zbl 0449.58019
[21] Kashiwara, M.; Kawai, T., Second microlocalization and asymptotic expansions, Lecture notes in physics, 126, (1980), Springer-Verlag Berlin, p. 21-76
[22] Laurent, Y., Polygône de Newton etb, Ann. sci. école norm. sup., 20, 391-441, (1987) · Zbl 0646.58021
[23] Laurent, Y.; Fernandes, T.Monteiro, Systèmes différentiels fuchsiens le long d’une sous-variété, Publ. RIMS Kyoto univ., 24, 397-431, (1988) · Zbl 0704.35032
[24] Laurent, Y.; Schapira, P., Images inverses des modules différentiels, Compositio math., 61, 229-251, (1987) · Zbl 0617.32014
[25] Maisonobe, Ph., \(D\)-modules: an overview towards effectivity, Computer algebra and differential equations, (1994), Cambridge Univ. Press Cambridge, p. 21-55 · Zbl 0804.35009
[26] Malgrange, B., Le polynôme de Bernstein d’une singularité isolée, Lecture notes in math., 459, (1975), Springer-Verlag Berlin, p. 98-119
[27] Malgrange, B., Polynômes de Bernstein-Sato et cohomologie évanescente, Astérisque, 101/102, 243-267, (1983) · Zbl 0528.32007
[28] Maynadier, H., Équations fonctionnelles pour une intersection complète quasi-homogène à singularité isolée, C.R. acad. sci. Paris, 322, 655-658, (1996) · Zbl 0849.32024
[29] M. Noro, T. Takeshima, 1992, Risa/Asir—A Computer Algebra System, Proceedings of International Symposium on Symbolic and Algebraic Computation, Paul, S. Wang, 387, 396, ACM, New York · Zbl 0757.13013
[30] Oaku, T., Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Jpn. J. indust. appl. math., 11, 485-497, (1994) · Zbl 0811.35006
[31] T. Oaku, 1994, Algorithms for finding the structure of solutions of a system of linear partial differential equations, Proceedings of International Symposium on Symbolic and Algebraic Computation, J. GathenM. Giesbrecht, 216, 223, ACM, New York · Zbl 0964.35505
[32] Oaku, T., Algorithmic methods for Fuchsian systems of linear partial differential equations, J. math. soc. jpn., 47, 297-328, (1995) · Zbl 0847.35032
[33] Oaku, T., An algorithm of computingb, Duke math. J., 87, (1997) · Zbl 0893.32009
[34] Oaku, T., Algorithms for thebD, J. pure appl. algebra, 117, (1997)
[35] Sabbah, C., Proximité évanescente. II. équations fonctionelles pour plusieurs fonctions analytiques, Compositio math., 64, 213-241, (1987) · Zbl 0632.32006
[36] Saito, M.; Takayama, N., Restrictions ofA1n-hypergeometric function, Int. J. math., 5, 537-560, (1994) · Zbl 0807.33010
[37] M. Saito, B. Sturmfels, N. Takayama, 1996, Integer programming and hypergeometric series · Zbl 1075.33504
[38] Sato, M.; Kashiwara, M.; Kumura, T.; Oshima, T., Micro-local analysis of prehomogeneous vector spaces, Invent math., 62, 117-179, (1980) · Zbl 0456.58034
[39] Shimoyama, T.; Yokoyama, K., Localization and primary decomposition of polynomial ideals, J. symbolic computation, 22, 247-277, (1996) · Zbl 0874.13022
[40] Takayama, N., Gröbner basis and the problem of contiguous relations, Japan J. appl. math., 6, 147-160, (1989) · Zbl 0691.68032
[41] N. Takayama, 1990, An algorithm of constructing the intergral of a module—an infinite dimensional analog of Gröbner basis, Proceedings of International Symposium on Symbolic and Algebraic Computation, S. WatanabeM. Nagata, 206, 211, ACM, New York
[42] Takayama, N., Computational algebraic analysis and connection formula, Surikaiseki kenkyusho Kôkyuroku, 811, 82-97, (1992)
[43] N. Takayama, 1991— Kan: A system for computation in algebraic analysis
[44] Yano, T., On the theory ofb, Publ. RIMS, Kyoto univ., 14, 111-202, (1978) · Zbl 0389.32005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.