Katsusuke Nabeshima
ABSTRACT
An algorithm for computing comprehensive Gröbner systems (CGS) is introduced in rings of linear partial differential operators. Their applications to b-functions are considered. The resulting algorithm designed for a wide use of computing comprehensive Gröbner systems can be used to compute all the roots of b-functions and relevant holonomic D-modules. Furthermore, with our implementation, effective methods are illustrated for computing holonomic D-modules associated with hypersurface singularities. It is shown that the proposed algorithm is full of versatility.
- D. Andres, V. Levandovskyy and J. Martín-Morales, Principal intersection and Bernstein-Sato polynomial of an affine variety. Proc. ISSAC2009, pages 231--238. ACM, 2009. Google Scholar
Digital Library
- J. Briançon and P. Maisonobe, Remarques sur l'idéal de Bernstein associé à des polynômes. prépublication Univ. Nice-Sophia Antipolis, nO 650, Mai, 2002.Google Scholar
- J. Bueso, J. Castro, J. Gómez-Torrecillas and J. Lobillo, An introduction to effective calculus in quantum groups. Rings, Hopf Algebras and Brauer Groups. Lecture Note in Pure and Applied Mathematics, 197, pages 55--83, Marcel-Dekker, New York, 1998.Google Scholar
- W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0--2 A computer algebra system for polynomial computations. 2015. http://www.singular.uni-kl.deGoogle Scholar
- D. Grayson and M. Stilman, Macaulay 2: a software system for algebraic geometry. http://www.math.uiuc.edu/Macaulay2Google Scholar
- M. Kalkbrener, On the stability of Gröbner bases under specializations. J. Symb. Comp., 24, pages 51--58, 1997. Google ScholarDigital Library
- A. Kandri-Rody and V. Weispfenning, Noncommutative Gröbner bases in algebras of solvable type. J. Symb. Comp., 9, pages 1--26, 1990. Google ScholarDigital Library
- D. Kapur, Y. Sun and D. Wang, A new algorithm for computing comprehensive Gröbner systems. Proc. ISSAC2010, pages 29--36. ACM, 2010. Google ScholarDigital Library
- M. Kashiwara, On the maximally overdetermined system of linear differential equations. Publ. Res. Inst. Math. Sci., 10, pages 563--579. 1974--1975.Google ScholarCross Ref
- M. Kashiwara, B-functions and holonomic systems: Rationality of roots of b-functions. Invent. Math., 38, pages 33--53. 1976.Google Scholar
- V. Levandovskyy, Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation. Doctoral Thesis, Universitat Kaiserslautern (Germany), 2005. https://kluedo.ub.uni-kl.de/files/1670/diss-web.pdfGoogle Scholar
- V. Levandovskyy and J. Martín-Morales, Computational D-module theory with SINGULAR, comparison with other systems and two new algorithms. Proc. ISSAC2008, pages 173--180. ACM, 2008. Google ScholarDigital Library
- V. Levandovskyy and J. Martín-Morales, Algorithms for checking rational roots of b-functions and their applications. J. Algebra, 352, pages 408--429. 2012.Google Scholar
- A. Leykin, Constructibility of the set of polynomials with a fixed Bernstein-Sato polynomial: an algorithmic approach J. Symb. Comp., 32, pages 663--675. 2001. Google ScholarDigital Library
- A. Montes and M. Wibmer, Gröbner bases for polynomial systems with parameters. J. Symb. Comp., 45, pages 1391--1425, 2010. Google ScholarDigital Library
- K. Nabeshima, Comprehensive Gröbner bases in various domains. Doctoral Thesis, Johannes Kepler Universitat Linz (Austria), 2007. http://www.risc.jku.at/publications/download/risc_3116/Nabeshima_thesis.pdfGoogle Scholar
- K. Nabeshima, Stability conditions of monomial bases and comprehensive Gröbner systems. Proc. CASC2012, LNCS, 7442, pages 248--259, Springer, 2012. Google ScholarDigital Library
- K. Nishiyama and M. Noro, Stratification associated with local b-functions. J. Symb. Comp., 45, pages 462--480. 2010. Google ScholarDigital Library
- M. Noro, An efficient modular algorithm for computing the global b-function. Proc. Mathematical Software, pages 147--157, World Scientific, 2002.Google ScholarCross Ref
- M. Noro and T. Takeshima, Risa/Asir- A computer algebra system. Proc. ISSAC1992, pages 387--396, ACM, 1992.\http://www.math.kobe-u.ac.jp/Asir/asir.html Google ScholarDigital Library
- T. Oaku and T. Shimoyama, A Gröbner basis method for modules over rings of differential operators, J. Symb. Comp., 18, pages 223--248. 1994. Google ScholarDigital Library
- T. Oaku, Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Japan J. Indust. Appl. Math., 11, pages 485--497. 1994.Google ScholarCross Ref
- T. Oaku, An algorithm of computing b-functions. Duke Math. J., 87, pages 115--132. 1997.Google Scholar
- T. Oaku, Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Adv. Appl. Math., 19, pages 61--105. 1997. Google ScholarDigital Library
- T. Oaku, Algorithms for the b-function and $D$-modules associated with a polynomial. J. Pure Appl. Algebra, 117 and 118, pages 495--518. 1997.Google Scholar
- OpenXM committers, OpenXM, a project to integrate mathematical software systems. 1998--2016. http://www.openxm.orgGoogle Scholar
- M. García Román and S. García Román, Gröbner bases and syzygies on bimodules over PBW algebras. J. Symb. Comp., 40, pages 1039--1052. 2005. Google ScholarDigital Library
- C. Sabbah, Introduction to the theory of $D$-modules. Lecture Notes, Nankai, 2011. http://www.math.polytechnique.fr/cmat/sabbah/sabbah_nankai110705.pdfGoogle Scholar
- A. Suzuki and Y. Sato, A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. Proc. ISSAC2006, pages 326--331, ACM, 2006. Google ScholarDigital Library
- S. Tajima, On b-functions and algebraic local cohomology classes attached to hypersurface with line singularities. RIMS Kkyroku Bessatsu, 52, pages 174--191. 2014.Google Scholar
- S. Tajima and Y. Umeta, Computing structures of holonomic $D$-modules associated with a simple line singularity. To appear in RIMS Kkyroku Bessatsu.Google Scholar
- J. Ucha and F. Castro-Jiménez, On the computation of Bernstein-Sato ideals. J. Symb. Comp., 37, pages 629--639. 2004.Google ScholarCross Ref
- V. Weispfenning, Comprehensive Gröbner bases. J. Symb. Comp., 14, pages 1--29. 1992. Google ScholarDigital Library
- T. Yano, On the holonomic system of $f^s$ and b-functions. Publ. RIMS, 12, pages 469--480. 1978.Google ScholarCross Ref
- T. Yano, On the theory of b-functions. Publ. RIMS, 14, pages 111--202. 1978.Google ScholarCross Ref
Index Terms
- Comprehensive Gröbner Systems in Rings of Differential Operators, Holonomic D-modules and B-functions
Recommendations
Comprehensive Gröbner systems in PBW algebras, Bernstein–Sato ideals and holonomic D-modules
AbstractA computation method of comprehensive Gröbner systems is introduced in Poincaré–Birkhoff–Witt (PBW) algebras and applications to Bernstein–Sato ideals, holonomic D-modules for parametric cases are considered. The proposed method ...
Gröbner bases and logarithmic D-modules
Let C[x]=C[x"1,...,x"n] be the ring of polynomials with complex coefficients and A"n the Weyl algebra of order n over C. Elements in A"n are linear differential operators with polynomial coefficients. For each polynomial f, the ring M=C[x]"f of rational ...
Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously
ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computationIn Kapur et al (ISSAC, 2010), a new method for computing a comprehensive Grobner system of a parameterized polynomial system was proposed and its efficiency over other known methods was effectively demonstrated. Based on those insights, a new approach ...
Comments