# 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 $$({\mathcal O}_X [s,f^{-1}]f^s) \otimes_{{\mathcal O}_X}{\mathcal M}$$”.

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
##### Keywords:
algebraic $$D$$-modules; $$b$$-function; algorithms
Full Text:
##### References:
  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  Becker, T.; Weispfenning, V., Gröbner bases, (1993), Springer-Verlag Berlin  Björk, J.E., Rings of differential operators, (1979), North-Holland Amsterdam  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  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  Borel, A., AlgebraicD, (1987), Academic Press Boston · Zbl 0642.32001  Buchberger, B., Ein algorithmisches kriterium für die Lösbarkeit eines algebraischen gleichungssystems, Aequationes math., 4, 374-383, (1970) · Zbl 0212.06401  Castro, F., Calculs effectifs pour LES idéaux d’operateurs différentiels, Travaux en cours, (1987), Hermann Paris, p. 1-19 · Zbl 0633.13009  Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms, (1992), Springer-Verlag Berlin  Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (1995), Springer-Verlag New York · Zbl 0819.13001  Eisenbud, D.; Huneke, C.; Vasconcelos, W., Direct methods for primary decomposition, Invent. math., 110, 207-235, (1992) · Zbl 0770.13018  Gailligo, A., Some algorithmic questions on ideals of differential operators, Lecture notes in computer science, 204, (1985), Springer-Verlag Berlin, p. 413-421  Gelfand, I.M.; Zelevinsky, A.V.; Kapranov, M.M., Hypergeometric functions and toric manifolds, Funct. anal. appl., 23, 94-106, (1989) · Zbl 0721.33006  Ginsburg, V., Characteristic varieties and vanishing cycles, Invent math., 84, 327-402, (1986) · Zbl 0598.32013  Hartshorne, R., Residues and duality, Lecture notes in mathematics, 20, (1966), Springer-Verlag Berlin  Kashiwara, M., Bb, Invent. math., 38, 33-53, (1976)  Kashiwara, M., On the holonomic systems of linear differential equations. II, Invent. math., 49, 121-135, (1978) · Zbl 0401.32005  Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations, Lecture notes in math., 1016, (1983), Springer-Verlag Berlin, p. 134-142  Kashiwara, M., Systems of microdifferential equations, (1983), Birkhäuser Boston · Zbl 0566.32022  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  Kashiwara, M.; Kawai, T., Second microlocalization and asymptotic expansions, Lecture notes in physics, 126, (1980), Springer-Verlag Berlin, p. 21-76  Laurent, Y., Polygône de Newton etb, Ann. sci. école norm. sup., 20, 391-441, (1987) · Zbl 0646.58021  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  Laurent, Y.; Schapira, P., Images inverses des modules différentiels, Compositio math., 61, 229-251, (1987) · Zbl 0617.32014  Maisonobe, Ph., $$D$$-modules: an overview towards effectivity, Computer algebra and differential equations, (1994), Cambridge Univ. Press Cambridge, p. 21-55 · Zbl 0804.35009  Malgrange, B., Le polynôme de Bernstein d’une singularité isolée, Lecture notes in math., 459, (1975), Springer-Verlag Berlin, p. 98-119  Malgrange, B., Polynômes de Bernstein-Sato et cohomologie évanescente, Astérisque, 101/102, 243-267, (1983) · Zbl 0528.32007  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  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  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  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  Oaku, T., Algorithmic methods for Fuchsian systems of linear partial differential equations, J. math. soc. jpn., 47, 297-328, (1995) · Zbl 0847.35032  Oaku, T., An algorithm of computingb, Duke math. J., 87, (1997) · Zbl 0893.32009  Oaku, T., Algorithms for thebD, J. pure appl. algebra, 117, (1997)  Sabbah, C., Proximité évanescente. II. équations fonctionelles pour plusieurs fonctions analytiques, Compositio math., 64, 213-241, (1987) · Zbl 0632.32006  Saito, M.; Takayama, N., Restrictions ofA1n-hypergeometric function, Int. J. math., 5, 537-560, (1994) · Zbl 0807.33010  M. Saito, B. Sturmfels, N. Takayama, 1996, Integer programming and hypergeometric series · Zbl 1075.33504  Sato, M.; Kashiwara, M.; Kumura, T.; Oshima, T., Micro-local analysis of prehomogeneous vector spaces, Invent math., 62, 117-179, (1980) · Zbl 0456.58034  Shimoyama, T.; Yokoyama, K., Localization and primary decomposition of polynomial ideals, J. symbolic computation, 22, 247-277, (1996) · Zbl 0874.13022  Takayama, N., Gröbner basis and the problem of contiguous relations, Japan J. appl. math., 6, 147-160, (1989) · Zbl 0691.68032  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  Takayama, N., Computational algebraic analysis and connection formula, Surikaiseki kenkyusho Kôkyuroku, 811, 82-97, (1992)  N. Takayama, 1991— Kan: A system for computation in algebraic analysis  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.