The F5 criterion revised

The purpose of this work is to generalize part of the theory behind Faugere's "F5" algorithm. This is one of the fastest known algorithms to compute a Grobner basis of a polynomial ideal I generated by polynomials f(1), ... ,f(m). A major reason for this is what Faugere called the algorithm's "new" criterion, and we call "the F5 criterion": it provides a sufficient condition for a set of polynomials G to be a Grobner basis. However. the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting: &-Grobner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f(1), ... ,f(m) be a regular sequence. (C) 2011 Elsevier Ltd. All rights reserved.