Signature-based algorithms to compute Groebner bases

TL;DR

This paper introduces a signature-based Buchberger-style algorithm for computing Groebner bases, unifying recent algorithms under this framework and analyzing their strategies both theoretically and empirically.

Contribution

It presents a new signature-based algorithm for Groebner bases and shows how existing algorithms fit into this framework, offering new insights into their strategies.

Findings

Different strategies significantly affect algorithm performance

Some strategies outperform others in empirical tests

Theoretical analysis reveals surprising results about strategy efficiency

Abstract

This paper describes a Buchberger-style algorithm to compute a Groebner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.