On the use of Buchberger criteria in G2V algorithm for calculating Gröbner bases

It has been experimentally demonstrated by Faugère that his F5 algorithm is the fastest algorithm for calculating Gröbner bases. Computational efficiency of F5 is due to not only applying linear algebra but also using the new F5 criterion for revealing useless zero reductions. At the ISSAC 2010 conference, Gao, Guan, and Volny presented G2V, a new version of the F5 algorithm, which is simpler than the original version of the algorithm. However, the incremental structure of G2V used in the algorithm for applying the F5 criterion is a serious obstacle from the point of view of application of Buchberger’s second criterion. In this paper, a modification of the G2V algorithm is presented, which makes it possible to use both Buchberger criteria. To experimentally study computational effect of the proposed modification, we implemented the modified algorithm in Maple. Results of comparison of G2V and its modified version on a number of test examples are presented.

[1]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[2]  Lei Huang,et al.  A new conception for computing gröbner basis and its applications , 2010, ArXiv.

[3]  Gwénolé Ars.  Applications des bases de Gröbner à la cryptograhie , 2005 .

[4]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[5]  Amir Hashemi,et al.  Extended F5 criteria , 2010, J. Symb. Comput..

[6]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[7]  Jee Koh,et al.  On efficient computation of grobner bases , 2008 .

[8]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[9]  Vladimir P. Gerdt,et al.  Involutive Algorithms for Computing Groebner Bases , 2005 .

[10]  Amir Hashemi,et al.  Applying IsRewritten criterion on Buchberger algorithm , 2011, Theor. Comput. Sci..

[11]  Shuhong Gao,et al.  A New Algorithm for Computing Grobner Bases , 2010 .

[12]  Bruno Buchberger,et al.  A criterion for detecting unnecessary reductions in the construction of Groebner bases , 1979, EUROSAM.

[13]  Christian Eder,et al.  F5C: A variant of Faugère's F5 algorithm with reduced Gröbner bases , 2009, J. Symb. Comput..

[14]  Christian Eder,et al.  Signature-based algorithms to compute Gröbner bases , 2011, ISSAC '11.

[15]  Svetlana Cojocaru,et al.  Computational Commutative and Non-Commutative Algebraic Geometry , 2005 .

[16]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[17]  Teo Mora,et al.  Solving Polynomial Equation Systems II: Macaulay's Paradigm and Gröbner Technology , 2005 .

[18]  Shuhong Gao,et al.  A new incremental algorithm for computing Groebner bases , 2010, ISSAC.

[19]  Rüdiger Gebauer,et al.  On an Installation of Buchberger's Algorithm , 1988, J. Symb. Comput..

[20]  B. Buchberger,et al.  Gröbner bases and applications , 1998 .