Predicting zero reductions in Gr\"obner basis computations

TL;DR

This paper compares various criteria for predicting zero reductions in Gr"obner basis computations, providing theoretical insights and practical evaluations to improve understanding of algebraic structures involved.

Contribution

It offers a detailed comparison of existing zero reduction prediction methods and introduces new insights into the algebraic structures underlying Gr"obner basis computations.

Findings

Signature-based criteria outperform classical criteria in regular sequences

Trade-offs identified between theoretical advantages and practical performance

New insights into syzygies and algebraic structures underlying zero reductions

Abstract

Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965 many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of Gebauer and M\"oller. A relatively new approach are signature-based criteria which were first used in Faug\`ere's F5 algorithm in 2002. For regular input sequences these criteria are known to compute no zero reduction at all. In this paper we give a detailed discussion on zero reductions and the corresponding syzygies. We explain how the different methods to predict them compare to each other and show advantages and drawbacks in theory and practice. With this a new insight into algebraic structures underlying Gr\"obner bases and their computations might be achieved.