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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:
 [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
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