A Signature-based Algorithm for computing Computing Gr\"obner Bases over Principal Ideal Domains

TL;DR

This paper introduces a signature-based algorithm for computing Gr"obner bases over principal ideal domains, extending existing methods from fields to more general rings, with proofs of correctness and initial experimental validation.

Contribution

It presents the first proof-of-concept signature-based algorithm for Gr"obner bases over integral domains, adapting M"oller's algorithm and ensuring correctness over PIDs.

Findings

Algorithm performs reductions with non-decreasing signatures.

Correctness and termination are proven for PIDs.

Initial experiments suggest potential applicability to UFDs.

Abstract

Signature-based algorithms have become a standard approach for Gr\"obner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gr\"obner bases over commutative integral domains. It is adapted from a general version of M\"oller's algorithm (1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).