Signature-based M\"oller's algorithm for strong Gr\"obner bases over PIDs

Abstract

Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several authors. In this paper, we present a signature-based version of M\"oller's classical variant of Buchberger's algorithm for computing strong Gr\"obner bases over Principal Ideal Domains (or PIDs). It ensures that the signatures do not decrease during the algorithm, which makes it possible to apply classical signature criteria for further optimization. In particular, with the F5 criterion, the signature version of M\"oller's algorithm computes a Gr\"obner basis without reductions to zero for a polynomial system given by a regular sequence. We also show how Buchberger's chain criterion can be implemented so as to be compatible with the signatures. We prove correctness and termination of the algorithm. Furthermore, we have written a toy implementation in Magma, allowing us to quantitatively compare the efficiency of the various criteria for eliminating S-pairs.