The F5 Criterion revised

TL;DR

This paper generalizes the F5 algorithm's theory by introducing new concepts like S-Groebner bases, leading to a simplified algorithm with broader applicability and improved understanding of the F5 criterion.

Contribution

It introduces S-Groebner bases and primitive S-irreducible polynomials, providing a revised F5 criterion and a simpler, more general algorithm for computing Groebner bases.

Findings

Proposed a new, simplified algorithm based on the revised F5 criterion.

Removed restrictions such as the need for the input polynomials to form a regular sequence.

Proved termination of the algorithm without previous constraints.

Abstract

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 Groebner 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 Groebner 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: S-Groebner 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.