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Comprehensive Gröbner systems in PBW algebras, Bernstein-Sato ideals and holonomic \(D\)-modules. (English) Zbl 1428.13047
Let \(f_1, \cdots, f_q\) be elements in the polynomial ring \(K[x,u]=K[x_1, \cdots , x_n,u]\) and \(A_n[s,u] = A_n \otimes_K K[s,u]\) be the Weyl algebra over \(K[s,u]=K[s_1, \cdots, s_q]\), where \(A_n\) is the Weyl algebra over K. Here \(u\) is as a parameter. The symbol \(f^s=f_1^{s_1} \cdots f_q^{s_q}\) is a generator of the \(A_n[s,u]\)-modul \(A_n[s] f^s\), where the action of differential operators on \(f^s\) is the intuitive one. One is interested in the annihilator \({\operatorname Ann}(f^s)\subset A_n[s]\) and also certain so-called Berstein-Sato ideals \[ \mathcal B_i =\{ b\in K[s,u] |b f^s \in A_n[s]\cdot f_if^s \}, \quad \mathcal B_\Sigma =\{ b\in K[s,u] | bf^s \in A_n[s]\cdot \sum_{i=1}^qf_if^s \}. \] A specialisation \(s\) is a homomorphism \(s: K[x,u]\to K'[x]\), wher \(K'\) is a field, which then also induces a homomorphism \(s: A_n[s,u]\to A_n'[s]\), where \(A_n'\) is the Weyl algebra over \(K'\). A finite set \(G\) in \(K[x,u]\) is a comprehenseve Gröbner basis if \(s(G)\) is a Gröbner basis for each specialisation \(s\). This notion was studied by V. Weispfenning [J. Symb. Comput. 14, No. 1, 1–29 (1992; Zbl 0784.13013)]. For the construction one decomposes \(\operatorname {Spec}K[u]\) into constructible subsets, where one provides a comprehensive Gröbner basis for one such subset at a time. In the paper this notion is considered for the non-commutative ring \(A_n[s,u] \) by extending algorithms by M. Kalkbrener [J. Symb. Comput. 24, No. 1, 51–58 (1997; Zbl 1054.13502)] in the commutative case to the ring \(A_n[s,u]\). This is implemented to compute comprehensive Gröbner bases of \({\operatorname {Ann}}(f^s)\) and also comprehensive Gröbner bases of \(\mathcal B_i\) and \(\mathcal B_\Sigma\). Some concrete examples are given and a link to used software is provided.
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D45 Local cohomology and commutative rings
13J05 Power series rings
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32A27 Residues for several complex variables
32C37 Duality theorems for analytic spaces
Full Text: DOI
[1] Aleksandrov, A. G., Normal forms of one-dimensional quasihomogeneous complete intersections, Math. USSR Sb., 45, 1, 1-30, (1983) · Zbl 0548.14017
[2] Andres, D.; Brickenstein, M.; Levandovskyy, V.; Martín-Morales, J.; Schönemann, H., Constructive D-module theory with singular, Math. Comput. Sci., 4, 359-383, (2010) · Zbl 1217.13011
[3] Apel, J., Gröbnerbasen in nichtkommutativen algebren und ihre anwendung, (1988), Universität Leipzig, Dissertation · Zbl 0716.16001
[4] Bahloul, R., Algorithm for computing Bernstein-Sato ideals associated with a polynomial mapping. effective methods in rings of differential operators, J. Symb. Comput., 32, 643-662, (2001) · Zbl 1035.16017
[5] Bahloul, R.; Oaku, T., Local Bernstein-Sato ideals: algorithm and examples, J. Symb. Comput., 45, 46-59, (2010) · Zbl 1184.14030
[6] Briançon, J., Maisonobe, P., 2002. Remarques sur l’idéal de Bernstein associé à des polynômes, prépublication, Univ. Nice-Sophia Antipolis, n° 650, Mai.
[7] Bueso, J.; Gómez-Torrecillas, J.; Lobillo, F., Re-filtering and exactness of the Gelfand-Kirillov dimension, Bull. Sci. Math., 125, 689-715, (2001) · Zbl 1006.16023
[8] Bueso, J.; Gómez-Torrecillas, J.; Verschoren, A., Algorithmic methods in non-commutative algebra, applications to quantum groups, (2003), Kluwer Academic Publishers · Zbl 1063.16054
[9] Cassou-Nogués, P., Racines de polynômes de Bernstein, Ann. Inst. Fourier, 36, 1-30, (1986) · Zbl 0597.32004
[10] Cassou-Nogués, P., Etude du comportement du polyôme de Bernstein lors d’une déformation à μ-constant de \(x^a + y^b\) avec \((a, b) = 1\), Compos. Math., 63, 291-313, (1987) · Zbl 0624.32006
[11] Cassou-Nogués, P., Polyôme de Bernstein générique, Abh. Math. Semin. Univ. Hamb., 58, 103-124, (1988) · Zbl 0685.32010
[12] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H., Singular 4-0-2 — a computer algebra system for polynomial computations, (2012)
[13] Grayson, D.; Stilman, M., Macaulay 2: a software system for algebraic geometry, (1992)
[14] Greuel, G.-M.; Levandovskyy, V.; Motsak, A.; Schönemann, H., PLURAL. A \scsingular 4.0 subsystem for computation with non-commutative polynomial algebras, (2015), Center for Computer Algebra, TU Kaiserslautern
[15] Guimarães, A.; Hefez, A., Bernstein-Sato polynomials and spectral numbers, Ann. Inst. Fourier, 57, 2031-2040, (2007) · Zbl 1130.32013
[16] Gyoja, A., Bernstein-Sato’s polynomial for several analytic functions, J. Math. Kyoto Univ., 33, 399-411, (1993) · Zbl 0797.32007
[17] Heinz, K., Solvable polynomial rings, (1992), Universität Passau, Doctoral Thesis
[18] Hertling, C.; Stahlke, C., Bernstein-Sato polynomials and tjurina number, Geom. Dedic., 75, 137-176, (1999) · Zbl 0955.32022
[19] Kalkbrener, M., On the stability of Gröbner bases under specializations, J. Symb. Comput., 24, 51-58, (1997) · Zbl 1054.13502
[20] Kandri-Rody, A.; Weispfenning, V., Non-commutative Gröbner bases in algebras of solvable type, J. Symb. Comput., 13, 117-131, (1990) · Zbl 0715.16010
[21] Kapur, D.; Sun, Y.; Wang, D., A new algorithm for computing comprehensive Gröbner systems, (Watt, S., International Symposium on Symbolic and Algebraic Computation, ISSAC, (2010), ACM), 29-36 · Zbl 1321.68533
[22] Kashiwara, M., b-functions and holonomic systems: rationality of roots of b-functions, Invent. Math., 38, 33-53, (1976) · Zbl 0354.35082
[23] Kato, M., The b-function of μ-constant deformation of \(x^7 + y^5\), Bull. Coll. Sci. Univ. Ryukyus, 32, 5-10, (1981) · Zbl 0496.32016
[24] Kato, M., The b-function of μ-constant deformation of \(x^9 + y^4\), Bull. Coll. Sci. Univ. Ryukyus, 33, 5-8, (1982) · Zbl 0505.32011
[25] Levandovskyy, V., Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementations, (2005), Universität Kaiserslautern, Doctoral Thesis
[26] Levandovskyy, V.; Martín-Morales, J., Computational D-module theory with singular, comparison with other systems and two new algorithms, (Jeffrey, D., International Symposium on Symbolic and Algebraic Computation, ISSAC2008, (2008), ACM), 173-180
[27] Levandovskyy, V.; Martín-Morales, J., Algorithms for checking rational roots of b-functions and their applications, J. Algebra, 352, 408-429, (2012) · Zbl 1276.32010
[28] Li, H., Noncommutative Gröbner bases and filtered-graded transfer, (2002), Springer · Zbl 1050.16022
[29] Maynadier, H., Polynôme de Bernstein-Sato associés à une intersection compleète quasi-homogène à singularité isolée, Bull. Soc. Math. Fr., 125, 547-571, (1997) · Zbl 0919.32022
[30] Miwa, T., Determination of \(b(s)\) - the case of quasi-homogeneous isolated singularity, RIMS Kokyuroku, 225, 62-71, (1975), (in Japanese)
[31] Montes, A.; Wibmer, M., Gröbner bases for polynomial systems with parameters, J. Symb. Comput., 45, 12, 1391-1425, (2010) · Zbl 1207.13018
[32] Nabeshima, K., Comprehensive Gröbner bases in various domains, (2007), Johannes Kepler Universität Linz, Doctoral Thesis
[33] Nabeshima, K., A speed-up of the algorithm for computing comprehensive Gröbner systems, (Brown, C., International Symposium on Symbolic and Algebraic Computation, ISSAC2007, (2007), ACM), 299-306 · Zbl 1190.13025
[34] Nabeshima, K., Stability conditions of monomial bases and comprehensive Gröbner systems, (Gerdt, V.; Koepf, W.; Mayr, E.; Vorozhtsov, E., Computer Algebra in Scientific Computing, CASC, Lecture Notes in Computer Science, vol. 7442, (2012), Springer), 248-259 · Zbl 1373.13030
[35] Nabeshima, K.; Ohara, K.; Tajima, S., Comprehensive Gröbner systems in rings of differential operators, holonomic D-modules and b-functions, (Rosenkranz, M., International Symposium on Symbolic and Algebraic Computation, ISSAC2016, (2016), ACM), 349-356 · Zbl 1360.13062
[36] Nabeshima, K.; Tajima, S., On efficient algorithms for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities and standard bases, (Nabeshima, K., International Symposium on Symbolic and Algebraic Computation, ISSAC2014, (2014), ACM), 351-358 · Zbl 1325.68297
[37] Nabeshima, K.; Tajima, S., Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals, J. Symb. Comput., 82, 91-122, (2017) · Zbl 1430.13026
[38] Noro, M.; Takeshima, T., Risa/asir - a computer algebra system, (Wang, P., International Symposium on Symbolic and Algebraic Computation, ISSAC1992, (1992), ACM), 387-396 · Zbl 0964.68597
[39] Oaku, T., An algorithm of computing b-functions, Duke Math. J., 87, 115-132, (1997) · Zbl 0893.32009
[40] 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
[41] Oaku, T., Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities, J. Symb. Comput., 50, 1-27, (2013) · Zbl 1284.68683
[42] Oaku, T.; Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation, J. Pure Appl. Algebra, 1139, 201-233, (1999) · Zbl 0960.14008
[43] Openxm committers, 1998-2017. openxm, a project to integrate mathematical software systems
[44] Sabbah, C., Proximité évanescente I. la structure polaire d’un D-module, Compos. Math., 62, 283-328, (1987) · Zbl 0622.32012
[45] Sabbah, C., Proximité évanescente II. équations fonctionnelles pour plusieurs fonctions analytiques, Compos. Math., 64, 213-241, (1987) · Zbl 0632.32006
[46] Saito, M., On the structure of Brieskorn lattice, Ann. Inst. Fourier, 39, 27-72, (1989) · Zbl 0644.32005
[47] Saito, M., Period mapping via Brieskorn modules, Bull. Soc. Math. Fr., 119, 141-171, (1991) · Zbl 0760.32009
[48] Saito, M., On b-function, spectrum and rational singularity, Math. Ann., 295, 51-74, (1993) · Zbl 0788.32025
[49] Saito, M., On microlocal b-function, Bull. Soc. Math. Fr., 122, 163-184, (1994) · Zbl 0810.32004
[50] Suzuki, A.; Sato, Y., A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases, (Dumas, J.-G., International Symposium on Symbolic and Algebraic Computation, (2006), ACM), 326-331 · Zbl 1356.13040
[51] Tajima, S., Local cohomology solutions of holonomic D-modules associated with a non-isolated hypersurface singularity, RIMS Kôkyûroku Bessatsu, (2018), to appear
[52] Tajima, S.; Nakamura, Y., Annihilating ideals for an algebraic local cohomology class, J. Symb. Comput., 44, 435-448, (2009) · Zbl 1171.32020
[53] Tajima, S.; Nakamura, Y., Algebraic local cohomology classes attached to unimodal singularities, Publ. Res. Inst. Math. Sci. Kyoto Univ., 48, 21-43, (2012) · Zbl 1246.14009
[54] Tajima, S.; Nakamura, Y.; Nabeshima, K., Standard bases and algebraic local cohomology for zero dimensional ideals, Adv. Stud. Pure Math., 56, 341-361, (2009) · Zbl 1194.13020
[55] Tajima, S.; Umeta, Y., Computing structures of holonomic D-modules associated with a simple line singularity, RIMS Kôkyûroku Bessatsu, 57, 125-140, (2016) · Zbl 1358.32017
[56] Ucha, J.; Castro-Jiménez, F., On the computation of Bernstein-Sato ideals, J. Symb. Comput., 37, 629-639, (2004) · Zbl 1137.16304
[57] Varchenko, A., Asymptotic Hodge structure on vanishing cohomology, Izv. Akad. Nauk USSR, Ser. Mat., 45, 540-591, (1981) · Zbl 0476.14002
[58] Weispfenning, V., Comprehensive Gröbner bases, J. Symb. Comput., 14, 1-29, (1992) · Zbl 0784.13013
[59] Yano, T., On the holonomic system of \(f^s\) and b-functions, Publ. RIMS, 12, 469-480, (1978) · Zbl 0389.32004
[60] Yano, T., On the theory of b-functions, Publ. RIMS, 14, 111-202, (1978) · Zbl 0389.32005
[61] Yoshinaga, E.; Suzuki, M., Normal forms of nondegenerate quasihomogeneous functions with inner modality ≤ 4, Invent. Math., 55, 185-206, (1979) · Zbl 0406.58008
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