Signature-based Möller's Algorithm for strong Gröbner bases over PIDs

Signature-based algorithms are the latest and most efficient approach as of today to compute Gröbner 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 Möller’s classical variant of Buchberger’s algorithm for computing strong Gröbner 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 Möller’s algorithm computes a Gröbner 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.