F4/5

TL;DR

This paper presents an algorithm combining F4-style reduction with F5 criteria for computing Gr"obner bases, offering a potentially more efficient approach with an available implementation in Sage.

Contribution

It introduces a novel algorithm that integrates F4 and F5 techniques for Gr"obner basis computation, with an open-source implementation and comparative analysis.

Findings

Algorithm effectively combines F4 and F5 methods.

Implementation available in Sage system.

Provides insights into algorithm efficiency and comparison.

Abstract

We describe an algorithm to compute Gr\"obner bases which combines F4-style reduction with the F5 criteria. Both F4 and F5 originate in the work of Jean-Charles Faug\`ere, who has successfully computed many Gr\"obner bases that were previously considered intractable. Another description of a similar algorithm already exists in Gwenole Ars' dissertation; unfortunately, this is only available in French, and although an implementation exists, it is not made available for study. We not only describe the algorithm, we also direct the reader to a study implementation for the free and open source Sage computer algebra system. We conclude with a short discussion of how the approach described here compares and contrasts with that of Ars' dissertation.