zbMATH — the first resource for mathematics

Stratification associated with local \(b\)-functions. (English) Zbl 1184.13079
Summary: The local \(b\)-function \(b_{f,p}(s)\) of an \(n\)-variate polynomial \(f \in \mathbb C[x](x=(x_{1},\dots ,x_n))\) at a point \(p\in \mathbb C^n\) is constant on each stratum of a stratification of \(\mathbb C^n\). We propose a new method for computing such a stratification and \(b_{f,p}(s)\) on each stratum. In the existing method proposed by T. Oaku [J. Pure Appl. Algebra 117–118, 495–518 (1997; Zbl 0918.32006)] a primary ideal decomposition of an ideal in \(\mathbb C[x,s]\) is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in \(\mathbb C[x]\) defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
16S32 Rings of differential operators (associative algebraic aspects)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
13N99 Differential algebra
Full Text: DOI
[1] Andres, D.; Levandovskyy, V.; Morales, J.M., Principal intersection and bernstein – sato polynomial of an affine variety, () · Zbl 1237.14004
[2] Assi, A.; Castro-Jiménez, F.J.; Granger, M., The Gröbner Fan of an \(A_n\)-module, J. pure appl. algebra, 150, 27-39, (2000) · Zbl 0967.32008
[3] Bahloul, R.; Oaku, T., Local bernstein – sato ideals: algorithm and examples, J. symbolic. comput., 45, 1, 46-59, (2010) · Zbl 1184.14030
[4] Bernstein, I.N., Modules over the ring of differential operators, Funct. anal. appl., 2, 1-16, (1971) · Zbl 0233.47031
[5] Briançon, J.; Granger, M.; Maisonobe, Ph.; Miniconi, M., Algorithme de calcul du polynôme de Bernstein: cas non dégénéré, Ann. inst. Fourier, 39, 553-610, (1989) · Zbl 0675.32008
[6] Briançon, J.; Maisonobe, Ph.; Merle, M., Constructibilité de l’ideal de Bernstein, Advanced studies in pure mathematics, 29, 79-95, (2000) · Zbl 1070.14507
[7] Briançon, J., Maisonobe, Ph., 2002. Remarques sur l’idéal de Bernstein associé à des polynômes, Preprint no. 650, Univ. Nice Sophia-Antipolis.
[8] Castro-Jiménez, F.J, Narváez-Macarro, L., 1997. Homogenising differential operators. Prepublicación de la Facultad de Matemáticas, Universidad de Sevilla, 36.
[9] Granger, M.; Oaku, T., Minimal filtered free resolutions and division algorithms for analytic \(D\)-modules, J. pure appl. algebra, 191, 1-2, 157-180, (2004) · Zbl 1064.16024
[10] Granger, M.; Oaku, T.; Takayama, N., Tangent cone algorithm for homogenized differential operators, J. comput., 39, 3-4, 417-431, (2005) · Zbl 1120.32300
[11] Greuel, G.-M., Pfister, G., Schönemann, H., 2001. Singular 3.0 — A Computer Algebra System for Polynomial Computations. In M. Kerber and M. Kohlhase: Symbolic Computation and Automated Reasoning, The Calculemus-2000 Symposium, 227-233.
[12] Greuel, G.-M.; Lossen, C.; Shustin, E., ()
[13] Kashiwara, M., \(B\)-functions and holonomic systems: rationality of roots of \(b\)-functions, Invent. math., 38, 33-53, (1976) · Zbl 0354.35082
[14] Leykin, A., Constructibility of the set of polynomials with a fixed bernstein – sato polynomial: an algorithmic approach, J. symbolic. comput., 32, 6, 663-675, (2001) · Zbl 1035.16018
[15] Levandovskyy, V.; Morales, J.M., Computational D-module theory with SINGULAR, comparison with other systems and two new algorithms, (), 173-180
[16] Malgrange, B., Intégrales asymptotiques et monodromie, Ann. sci. ENS, 4, 405-430, (1974) · Zbl 0305.32008
[17] Mebkhout, Z.; Narváez-Macarro, L., La théorie du polynôme de bernstein – sato pour LES algèbres de Tate et de dwork – monsky – washintzer, Ann. sci. école norm. sup. (4), 24, 2, 227-256, (1991) · Zbl 0765.14009
[18] Nakayama, H., Algorithm computing the \(b\) function by an approximate division algorithm in \(\hat{D}\), J. symbolic. comput., 44, 5, 449-462, (2009) · Zbl 1178.68684
[19] Noro, M.; Yokoyama, K., A modular method to compute the rational univariate representation of zero-dimensional ideals, J. symbolic. comput., 28, 1, 243-263, (1999) · Zbl 0945.13010
[20] Noro, M., An efficient modular algorithm for computing the global \(b\)-function, (), 147-157 · Zbl 1057.68149
[21] Noro, M., (), 147-162
[22] Oaku, T., An algorithm of computing \(b\)-functions, Duke math. J., 87, 115-132, (1997) · Zbl 0893.32009
[23] Oaku, T., Algorithms for \(b\)-functions, restrictions, and algebraic local cohomology groups of \(d\)-modules, Adv. appl. math., 19, 61-105, (1997) · Zbl 0938.32005
[24] Oaku, T., Algorithms for the \(b\)-function and \(D\)-modules associated with a polynomial, J. pure appl. algebra, 117 and 118, 495-518, (1997) · Zbl 0918.32006
[25] OpenXM committers, 1998-2009. OpenXM, a project to integrate mathematical software systems, http://www.openxm.org.
[26] Saito, M.; Sturmfels, B.; Takayama, N., Gröbner deformations of hypergeometric differential equations, () · Zbl 0946.13021
[27] Sato, M.; Kashiwara, M.; Kimura, T.; Oshima, T., Micro-local analysis of prehomogeneous vector spaces, Invent. math., 62, 117-179, (1980) · Zbl 0456.58034
[28] Schultze, M., Algorithms for the gauss – manin connection, J. symbolic. comput., 23, 5, 549-564, (2001) · Zbl 0987.32014
[29] Traverso, C., Groebner trace algorithms, (), 125-138
[30] Yano, T., On the holonomic system of \(f^s\) and \(b\)-functions, Publ. RIMS Kyoto univ., 12, Suppl., 469-480, (1978) · Zbl 0389.32004
[31] Yano, T., On the theory of \(b\)-functions, 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.