skip to main content
10.1145/1277548.1277589acmconferencesArticle/Chapter ViewAbstractPublication PagesissacConference Proceedingsconference-collections
Article

A speed-up of the algorithm for computing comprehensive Gröbner systems

Published: 29 July 2007 Publication History

Abstract

We introduce a new algorithm for computing comprehensive Gröbner systems.There exists the Suzuki-Sato algorithm for computing comprehensive Gröbner systems. The Suzuki-Sato algorithm often creates overmuch cells of the parameter space for comprehensive Gröbner systems. Therefore the computation becomes heavy. However, by using inequations ("not equal zero"), we can obtain different cells. In many cases, this number of cells of parameter space is smaller than that of Suzuki-Sato's. Therefore, our new algorithm is more efficient than Suzuki-Sato's one, and outputs a nice comprehensive Gröbner system. Our new algorithm has been implemented in the computer algebra system Risa/Asir We compare the runtime of our implementation with the Suzuki-Sato algorithm and find our algorithm superior in many cases.

References

[1]
Becker, T. On Gröbner bases under specialization. Applicable Algebra in Engineering, Communication and Computing 5:1--8, 1994.
[2]
Dolzmann, A. and Sturm, T. Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2):2--9, 1997.
[3]
Gianni, P. Properties of Gröbner bases under specializations. In Davenport, J., editor, EUROCAL'87 pages 293--297. ACM Press, 1987.
[4]
Kalkbrener, M. On the Stability of Gröbner Bases Under Specializations. Journal of Symbolic Computation 24:51--58, 1997.
[5]
Manubens, M. and Montes, A. Improving DISPGB algorithm using the discriminant ideal. Journal of Symbolic Computation 41:1245--1263, 2006.
[6]
Montes, A. A new algorithm for discussing Gröbner basis with parameters. Journal of Symbolic Computation 33/1-2:183--208, 2002.
[7]
Nabeshima, K. Reduced Gröbner bases in polynomial rings over a polynomial ring. In Wang, D. and Zheng, Z., editors, International Conference on Mathematical Aspects of Computer and Information Sciences pages 15--32, 2006.
[8]
Nabeshima, K. PGB: A Package for Computing Parametric Gröbner Bases and Related Objects. 2007. preprint.
[9]
Noro, M. and Takeshima, T. Risa/Asir-A Computer Algebra System. In Wang, P., editor, International Symposium on Symbolic and Algebraic Computation pages 387--396. ACM Press, 1992. http://www.math.kobe-u.ac.jp/Asir/asir.html
[10]
Suzuki, A. and Sato, Y. An alternative approach to Comprehensive Gröbner bases. Journal of Symbolic Computation 36/3-4:649--667, 2003.
[11]
Suzuki, A. and Sato, Y. A Simple Algorithm to compute Comprehensive Gröbner Bases using Gröbner bases. In Dumas, J-G., editor, International Symposium on Symbolic and Algebraic Computation pages 326--331. ACM Press, 2006.
[12]
Weispfenning, V. Comprehensive Gröbner bases. Journal of Symbolic Computation 14/1:1--29, 1992.
[13]
Weispfenning, V. Canonical Comprehensive Gröbner bases. In Mora, T., editor,International Symposium on Symbolic and Algebraic Computation pages 270--278. ACM Press, 2002.

Cited By

View all
  • (2024)A Parametric $ F_4 $ AlgorithmIranian Journal of Mathematical Sciences and Informatics10.61186/ijmsi.19.1.11719:1(117-133)Online publication date: 1-Apr-2024
  • (2024)An Algorithm for Computing Greatest Common Right Divisors of Parametric Ore PolynomialsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669694(226-233)Online publication date: 16-Jul-2024
  • (2024)Algorithmic Detection of Conserved Quantities for Finite-Difference SchemesMathematics in Computer Science10.1007/s11786-024-00595-w18:4Online publication date: 15-Nov-2024
  • Show More Cited By

Index Terms

  1. A speed-up of the algorithm for computing comprehensive Gröbner systems

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Conferences
    ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
    July 2007
    406 pages
    ISBN:9781595937438
    DOI:10.1145/1277548
    • General Chair:
    • Dongming Wang
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

    Sponsors

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 29 July 2007

    Permissions

    Request permissions for this article.

    Check for updates

    Author Tags

    1. Gröbner bases
    2. comprehensive Gröbner bases

    Qualifiers

    • Article

    Conference

    ISSAC07
    Sponsor:
    ISSAC07: International Symposium on Symbolic and Algebraic Computation
    July 29 - August 1, 2007
    Ontario, Waterloo, Canada

    Acceptance Rates

    Overall Acceptance Rate 395 of 838 submissions, 47%

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)2
    • Downloads (Last 6 weeks)0
    Reflects downloads up to 14 Feb 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2024)A Parametric $ F_4 $ AlgorithmIranian Journal of Mathematical Sciences and Informatics10.61186/ijmsi.19.1.11719:1(117-133)Online publication date: 1-Apr-2024
    • (2024)An Algorithm for Computing Greatest Common Right Divisors of Parametric Ore PolynomialsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669694(226-233)Online publication date: 16-Jul-2024
    • (2024)Algorithmic Detection of Conserved Quantities for Finite-Difference SchemesMathematics in Computer Science10.1007/s11786-024-00595-w18:4Online publication date: 15-Nov-2024
    • (2024)Improvement of an Incremental Signature-Based Comprehensive Gröbner System AlgorithmMathematics in Computer Science10.1007/s11786-024-00587-w18:2Online publication date: 27-Jun-2024
    • (2023)Generic Gröbner basis of a parametric ideal and its application to a comprehensive Gröbner systemApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-023-00620-835:1(55-70)Online publication date: 3-Aug-2023
    • (2022)Rational Univariate Representation of Zero-Dimensional Ideals with ParametersProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3535496(217-224)Online publication date: 4-Jul-2022
    • (2022)An Extended GCRD Algorithm for Parametric Univariate Polynomial Matrices and Application to Parametric Smith FormJournal of Symbolic Computation10.1016/j.jsc.2022.07.006Online publication date: Aug-2022
    • (2021)Comprehensive Characteristic Decomposition of Parametric Polynomial SystemsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465536(123-130)Online publication date: 18-Jul-2021
    • (2020)An extended GCD algorithm for parametric univariate polynomials and application to parametric smith normal formProceedings of the 45th International Symposium on Symbolic and Algebraic Computation10.1145/3373207.3404019(442-449)Online publication date: 20-Jul-2020
    • (2020)An Improvement of the Rational Representation for High-Dimensional SystemsJournal of Systems Science and Complexity10.1007/s11424-020-9316-4Online publication date: 7-Nov-2020
    • Show More Cited By

    View Options

    Login options

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Figures

    Tables

    Media

    Share

    Share

    Share this Publication link

    Share on social media