Abstract
Gao, Volny and Wang (2010) gave a simple criterion for signature-based algorithms to compute Gröbner bases. It gives a unified frame work for computing Gröbner bases for both ideals and syzygies, the latter is very important in free resolutions in homological algebra. Sun and Wang (2011) later generalized the GVW criterion to a more general situation (to include the F5 Algorithm). Signature-based algorithms have become increasingly popular for computing Gröbner bases. The current paper introduces a concept of factor pairs that can be used to detect more useless J-pairs than the generalized GVW criterion, thus improving signature-based algorithms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buchberger B, Ein Algorithmus zum auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomial, PhD thesis, 1965.
Faugere J C, A new effcient algorithm for computing Gröbner bases without reduction to zero (F5), Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, 2002, 75–83.
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.
Gao S H, Volny F, and Wang M S, A new algorithm for computing Gröbner bases, http://www.icar.org, 2010, 641.
Gao S H, Guan Y H, and Volny F, A new incremental algorithm for computing Gröbner bases, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 2010, 13–19.
Sun Y and Wang D K, The F5 algorithm in Buchberger’s style, Journal of Systems Science and Complexity, 2010, 24(6): 1218–1231.
Sun Y and Wang D K, A new proof for the correctness of the F5 algorithm. Preprint, arXiv: 1004.0084v4, 2010.
Sun Y and Wang D K, A generalized criterion for signature related Gröbner basis algorithms, Proceedings of ISSAC’11, ACM Press, New York, USA, 2011, 337–344.
Sun Y and Wang D K, Solving detachability problem for the polynomial ring by signature-based Gröbner basis algorithms, Preprint, arXiv: 1108.1301, 2011.
Zobnin A, Generalization of the F5 algorithm for calculating Gröbner bases for polynomial ideals, Programming and Computer Software, 2010, 36(2): 75–82.
Eisenbud D, Commutative Algebra with a View Toward Algebraic Geometry, Springer, 2004.
Buchberger B, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, 1979, 72: 3–21.
Courtois N, Klimov E, Patarin J, and Shamir A, Effcient algorithms for solving overdefned systems of multivariate polynomial equations, Advances in Cryptology, Eurocrypt 2000, LNCS 1807, Springer-Verlag, 2000, 1807: 392–407.
Faugere J C, A new effcient algorithm for computing Gröbner bases (F4), Journal of Pure and Applied Algebra, 1999, 139(1–3): 61–88.
Faugre J C and Joux A, Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases, Advances in Cryptology CRYPTO 2003, Springer, 2003, 44–60.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the National Natural Science Foundation of China under Grant Nos. 11471108, 11426101, Hunan Provincial Natural Science Foundation of China under Grant Nos. 14JJ6027, 2015JJ2051, and Fundamental Research Funds for the Central Universities of Central South University under Grant No. 2013zzts008.
This paper was recommended for publication by Editor HU Lei.
Rights and permissions
About this article
Cite this article
Zheng, L., Liu, J., Liu, W. et al. A new signature-based algorithms for computing Gröbner bases. J Syst Sci Complex 28, 210–221 (2015). https://doi.org/10.1007/s11424-015-2260-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-015-2260-z