Modifying Faug\`ere's F5 Algorithm to ensure termination

TL;DR

This paper analyzes the termination issues of Faugère's F5 algorithm for Gröbner basis computation and proposes three variants, including a new approach called F5+, to guarantee termination while maintaining efficiency.

Contribution

It provides a detailed analysis of the termination problem and introduces three variants, notably F5+, which uses a degree bound and critical pair distinction to ensure termination.

Findings

F5+ effectively guarantees termination.

F5+ is more efficient than previous variants.

Experimental results support F5+'s efficiency.

Abstract

The structure of the F5 algorithm to compute Gr\"obner bases makes it very efficient. However, while it is believed to terminate for so-called regular sequences, it is not clear whether it terminates for all inputs. This paper has two major parts. In the first part, we describe in detail the difficulties related to a proof of termination. In the second part, we explore three variants that ensure termination. Two of these have appeared previously only in dissertations, and ensure termination by checking for a Gr\"obner basis using traditional criteria. The third variant, F5+, identifies a degree bound using a distinction between "necessary" and "redundant" critical pairs that follows from the analysis in the first part. Experimental evidence suggests this third approach is the most efficient of the three.