On Two Signature Variants Of Buchberger's Algorithm Over Principal Ideal Domains

TL;DR

This paper explores two signature-based variants of Buchberger's algorithm for computing Gröbner bases over principal ideal domains, proving their correctness and comparing their performance.

Contribution

It introduces and analyzes two new signature-based algorithms for Gröbner bases over PIDs, extending existing methods from fields to more general rings.

Findings

Both algorithms are correct for PIDs.

Performance varies depending on the variant and implementation.

The algorithms provide additional data useful for syzygy modules and coefficient computation.

Abstract

Signature-based algorithms have brought large improvements in the performances of Gr\"obner bases algorithms for polynomial systems over fields. Furthermore, they yield additional data which can be used, for example, to compute the module of syzygies of an ideal or to compute coefficients in terms of the input generators. In this paper, we examine two variants of Buchberger's algorithm to compute Gr\"obner bases over principal ideal domains, with the addition of signatures. The first one is adapted from Kandri-Rody and Kapur's algorithm, whereas the second one uses the ideas developed in the algorithms by L. Pan (1989) and D. Lichtblau (2012). The differences in constructions between the algorithms entail differences in the operations which are compatible with the signatures, and in the criteria which can be used to discard elements. We prove that both algorithms are correct and discuss their relative performances in a prototype implementation in Magma.