Skip to main content
SpringerLink
Log in

GVW algorithm over principal ideal domains

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

GVW algorithm was given by Gao, Wang, and Volny in computing a Gröbner bases for ideal in a polynomial ring, which is much faster and more simple than F5. In this paper, the authors generalize GVW algorithm and present an algorithm to compute a Gröbner bases for ideal when the coefficient ring is a principal ideal domain.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Buchberger B and Winkler F, Edt., Gröbner Bases and Application, London Mathematical Society Lecture Note Series, 251, Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  2. Lazard D, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, EUROCAL’83: Proceedings of the European Computer Algebra Conference on Computer Algebra, London, UK, Springer-Verlag, 1983, 146–156.

    Google Scholar 

  3. Buchberger B, An algorithm for finding a bases for the redidue class ring of a zero dimensiomal polynomial, Ph. D. Thesis, Universität Innsbruck, Institut für Mathematik, 1965.

    Google Scholar 

  4. Buchberger B, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, EUROSAM’79: Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, 1979, 3–21.

    Chapter  Google Scholar 

  5. Buchberger B, Gröbner-Bases: An Algorithmic Method in Polynomial Ideal Theory, Reidel Publishing Company, Dodrecht Boston Lancaster, 1985.

    Google Scholar 

  6. Eder C and Perry J, F5C: A variant of Faugere’s F5 algorithm with reduced Gröbner bases, Journal of Symbolic Computation, 2009, 45(12): 1442–1458.

    Article  MathSciNet  Google Scholar 

  7. Faugere J C, A new effcient algorithm for computing Gröbner bases (F4), Journal of Pure and Applied Algebra, 1999, 139(1–3): 61–88.

    Article  MathSciNet  MATH  Google Scholar 

  8. Faugere J C, A new effcient algorithm for computing Gröbner bases without reduction to zero (F5), ISSAC’ 02: Proceedings of the 2002 international symposium on Symbolic and algebraic computation, 2002, 75–83.

    Chapter  Google Scholar 

  9. Sun Y and Wang D, F5B: A new proof of the F5 algorithm, CoRR, arXiv:1004.0084, 2010.

    Google Scholar 

  10. Gao S, Guan Y, and Volny F, A new incremental algorithm for computing Gröbner bases, ISSAC 10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 2010, 13–19.

    Chapter  Google Scholar 

  11. Gao S, Volny F, and Wang M, Gröbner-bases: A new algorithm for computing Gröbner bases, http://www.icar.org.641, 2010.

    Google Scholar 

  12. Lazard D, Revisiting strong Gröbner bases over Euclidean domains, http://library.wolfram.com/infocenter/MathSource/7522, 2003.

    Google Scholar 

  13. Graham H and Salagean A, Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Austral. Math. Soc. 2001, 64: 505–528.

    Article  MathSciNet  MATH  Google Scholar 

  14. Becker T and Weispfenning V, Gröbner Bases — A Computational Approach to Commutative Algebra, GTM 141, New York: Springer-Verlag, 1993.

    Google Scholar 

  15. Gert-Martin G and Gerhard P, A Singular Introduction to Commutative Algebra, New York, Spinger-Verlag, 2002.

    MATH  Google Scholar 

  16. Courtois N, Klimov E, Patarin J, and Shamir A, Effcient algorithms for solving overdefned systems of multivariate polynomial equations, Advances in Cryptology, Eurocrypt 2000, Lecture Notes in Computer Science, Springer-Verlag, 2000, 1807: 392–407.

    Google Scholar 

  17. Sun Y and Wang D, A generalized criterion for signature related Gröbner bases algorithms, CoRR, arXiv: 1101.3382, 2011.

    Google Scholar 

  18. Huang L, A new conception for computing Gröbner bases and its applications, CoRR, arXiv: 1012.5425v2, 2010.

    Google Scholar 

  19. Hashemi A and Ars G, Extended F5 criteria, Journal of Symbolic Computation, 2010, 45: 1330–1340.

    Article  MathSciNet  MATH  Google Scholar 

  20. Möller H M, Mora T, and Traverso C, Gröbner bases computation using syzygies, ISSAC’92: Papers from the international symposium on Symbolic and algebraic computation, New York, USA, ACM, 1992, 320–328.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan, 411201, China

    Dongmei Li & Jinwang Liu

  2. Department of Mathematical and Statistical, Central South University, Changsha, 410083, China

    Dongmei Li

  3. School of Mathematical and Statistical, Central South University, Changsha, 410083, China

    Weijun Liu & Licui Zheng

Authors
  1. Dongmei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinwang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weijun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Licui Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weijun Liu.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11071062, 11271208, and Scientific Research Fund of Hunan Province Education Department under Grant Nos. 10A033, 12C0130.

This paper was recommended for publication by Editor GAO Xiaoshan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, D., Liu, J., Liu, W. et al. GVW algorithm over principal ideal domains. J Syst Sci Complex 26, 619–633 (2013). https://doi.org/10.1007/s11424-013-2130-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-013-2130-5

Keywords

Search

Navigation