Effective computation of strong Gröbner bases over Euclidean domains

Abstract

Buchberger and, independently, Kandri-Rody and Kapur, defined a strong Gröbner basis for a polynomial ideal over a Euclidean domain in a way that gives rise to canonical reductions. This retains what is perhaps the most important property of Gröbner bases over fields. A difficulty is that these can be substantially harder to compute than their field counterparts. We extend their results for computing these bases to give an algorithm that is effective in practice. In particular, we show how to use S-polynomials (rather than “critical pairs”) so that the algorithm becomes quite similar to that for fields, and thus known strategies for the latter may be employed. We also show how Buchberger’s important criteria for detection of unneeded S-polynomials can be extended to work over Euclidean domains. We illustrate with some examples.