Comprehensive Gröbner bases and regular rings

Abstract

Commutative von Neumann regular rings can be viewed as certain subdirect products of fields. So in some sense they can code arbitrary sets of fields. It was shown in 1987 that most of the Gröbner basis theory over fields initiated by B. Buchberger can be extended to finitely generated ideals over commutative von Neumann regular rings. On the other hand, the construction of Comprehensive Gröbner Bases (CGBs) over fields shows that the Gröbner basis theory over fields can be extended to polynomials with parametric coefficients. Here we show that there is a surprisingly close relationship between comprehensive Gröbner bases over fields and non-parametric Gröbner bases over commutative von Neumann regular rings. Thus the latter can be viewed as an alternative to CGBs. Moreover we show that Gröbner bases over commutative von Neumann regular rings do in fact also cover parametric Gröbner bases over these rings. These facts also offer new algorithmic perspectives on parametric Gröbner bases. They form a strong generalization of the earlier results of Y. Sato and A. Suzuki.