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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.
##### MSC:
 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
##### Software:
Macaulay2; OpenXM; PGB; Plural; Risa/Asir; SINGULAR
Full Text:
##### References:
 [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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