Algorithmic stratification of \(\mathbb R\operatorname{Hom}_{\mathcal D}(\mathcal M, \mathcal N)\) for regular algebraic \(\mathcal D\)-modules on \(\mathbb C^n\). (English) Zbl 1130.32002
Let \(M\) be a holonomic \(\mathcal{D}\)-module on \(\mathbb{C}^n\). We give an algorithm to stratify \(\mathbb{C}^n\) such that on all strata \(X\) each restriction (derived inverse image) module \(H^i (\rho_{X, \mathbb{C}^n}(M))\) is a connection. For regular holonomic modules this stratifies the solution complex \(\mathbb{R}\operatorname{Hom}_{\mathcal{D}_{\mathbb{C}^{n, an}}} (M_{an}, \mathcal{O}_{an})\) of sheaves of (higher) holomorphic solutions of \(M\). We also give an algorithm to compute the dimension of the cohomology modules of the solution complex over all strata.
Reviewer: Gheorghe Gussi (Bucureşti)
MSC:
| 32C38 | Sheaves of differential operators and their modules, \(D\)-modules |
| 14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |
| 68W30 | Symbolic computation and algebraic computation |
References:
| [1] | Assi, A.; Castro-Jiménez, F. J.; Granger, J. M., How to calculate the slopes of a \(D\)-module, Compositio Math., 104, 2, 107-123 (1996) · Zbl 0862.32005 |
| [2] | Björk, J.-E., Analytic \(D\)-modules and Applications, Mathematics and its Applications, vol. 247 (1993), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0805.32001 |
| [3] | Borel, A.; Grivel, P.-P.; Kaup, B.; Haefliger, A.; Malgrange, B.; Ehlers, F., Algebraic \(D\)-modules (1987), Academic Press Inc.: Academic Press Inc. Boston, MA |
| [4] | Eisenbud, D.; Huneke, C.; Vasconcelos, W., Direct methods for primary decomposition, Invent. Math., 110, 2, 207-235 (1992) · Zbl 0770.13018 |
| [5] | Geĺfand, I. M.; Graev, M. I.; Zelevinskiĭ, A. V., Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR, 295, 1, 14-19 (1987) · Zbl 0661.22005 |
| [6] | Geĺfand, I. M.; Zelevinskiĭ, A. V.; Kapranov, M. M., Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen., 23, 2, 12-26 (1989) |
| [7] | Grayson, D., Stillman, M., \(1996. Macaulay 2 \mathcal{D}\) htttp://math.uiuc.edu; Grayson, D., Stillman, M., \(1996. Macaulay 2 \mathcal{D}\) htttp://math.uiuc.edu |
| [8] | Kashiwara, M., 1974-1975. On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. 10, 563-579; Kashiwara, M., 1974-1975. On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. 10, 563-579 · Zbl 0313.58019 |
| [9] | Kashiwara, M., On the holonomic systems of linear differential equations. II, Invent. Math., 49, 2, 121-135 (1978) · Zbl 0401.32005 |
| [10] | Malgrange, B., Le polynôme de Bernstein d’une singularité isolée, (Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974). Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Lecture Notes in Math., vol. 459 (1975), Springer: Springer Berlin), 98-119 · Zbl 0308.32007 |
| [11] | Ōaku, T., Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Japan J. Indust. Appl. Math., 11, 3, 485-497 (1994) · Zbl 0811.35006 |
| [12] | Oaku, T.; Takayama, N., Algorithms for \(D\)-modules—restriction, tensor product, localization, and local cohomology groups, J. Pure Appl. Algebra, 156, 2-3, 267-308 (2001) · Zbl 0983.13008 |
| [13] | Saito, M.; Sturmfels, B.; Takayama, N., Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, vol. 6 (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0946.13021 |
| [14] | Walther, U., 2004. Experiments with the restriction algorithm (preprint) pp. 1-5; Walther, U., 2004. Experiments with the restriction algorithm (preprint) pp. 1-5 |
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