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.