A new algorithm for computing Groebner bases

Buchberger’s algorithm for computing Gröbner bases was introduced in 1965, and subsequently there have been extensive efforts in improving its efficiency. Major algorithms include F4 (Faugère 1999), XL (Courtois et al. 2000) and F5 (Faugère 2002). F5 is believed to be the fastest algorithm known in the literature. Most recently, Gao, Guan and Volny (2010) introduced an incremental algorithm (G2V) that is two to ten times faster than F5. In this paper, a new algorithm is presented that matches Buchberger’s algorithm in simplicity and yet is more flexible than G2V. Given a list of polynomials, the new algorithm computes simultaneously a Gröbner basis for the ideal generated by the polynomials and a Gröbner basis for the leading terms of the syzygy module of the polynomials. For any term order for the ideal, one may vary the term order for the syzygy module. Under one term order for the syzygy module, the new algorithm specializes to the G2V algorithm, and under another term order for the syzygy module, the new algorithm may be several times faster than G2V, as indicated by computer experiments on benchmark examples.