skip to main content
research-article

Conditional GrÖbner Basis: GrÖbner Basis Detection with Parameters

Published:13 December 2023Publication History
Skip Abstract Section

Abstract

For a given finite polynomial set, finding a monomial order such that the given set is already a GrÖbner basis for the ideal generated by the given set with respect to the found monomial order is called GrÖbner basis detection (GBD) problem and there is also its simpler version, called structural GrÖbner basis detection (SGBD) problem. In this short communication, we give algorithms to solve these problems for polynomials with parameters on their coefficients.

References

  1. Peter Gritzmann and Bernd Sturmfels. Minkowski addition of polytopes: computational complexity and applications to GrÖbner bases. SIAM J. Discrete Math., 6(2):246--269, 1993.Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Amir Hashemi and Delaram Talaashrafi. A note on dynamic GrÖbner bases computation. In Computer algebra in scientific computing, volume 9890 of Lecture Notes in Comput. Sci., pages 276--288. Springer, Cham, 2016.Google ScholarGoogle ScholarCross RefCross Ref
  3. Michael Kalkbrener. On the stability of GrÖbner bases under specializations. J. Symbolic Comput., 24(1):51--58, 1997.Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Deepak Kapur, Yao Sun, and Dingkang Wang. A new algorithm for computing comprehensive GrÖbner systems. In ISSAC 2010---Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, pages 29--36. ACM, New York, 2010.Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Kosaku Nagasaka. Computing a structured GrÖbner basis approximately. In ISSAC 2011---Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, pages 273--280. ACM, New York, 2011.Google ScholarGoogle Scholar
  6. Lorenzo Robbiano. Term orderings on the polynomial ring. In EUROCAL '85, Vol. 2 (Linz, 1985), volume 204 of Lecture Notes in Comput. Sci., pages 513--517. Springer, Berlin, 1985.Google ScholarGoogle ScholarCross RefCross Ref
  7. Bernd Sturmfels and Markus Wiegelmann. Structural GrÖbner basis detection. Appl. Algebra Engrg. Comm. Comput., 8(4):257--263, 1997.Google ScholarGoogle ScholarCross RefCross Ref
  8. Volker Weispfenning. Comprehensive GrÖbner bases. J. Symbolic Comput., 14(1):1--29, 1992.Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Conditional GrÖbner Basis: GrÖbner Basis Detection with Parameters
      Index terms have been assigned to the content through auto-classification.

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Communications in Computer Algebra
        ACM Communications in Computer Algebra  Volume 57, Issue 3
        September 2023
        80 pages
        ISSN:1932-2240
        DOI:10.1145/3637529
        Issue’s Table of Contents

        Copyright © 2023 Copyright is held by the owner/author(s)

        Permission to make digital or hard copies of part or all 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 third-party components of this work must be honored. For all other uses, contact the Owner/Author.

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 13 December 2023

        Check for updates

        Qualifiers

        • research-article
      • Article Metrics

        • Downloads (Last 12 months)25
        • Downloads (Last 6 weeks)3

        Other Metrics

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader