Relative Gröbner and Involutive Bases for Ideals in Quotient Rings

We extend the concept of Grobner bases to relative Grobner bases for ideals in and modules over quotient rings of a polynomial ring over a field. We develop a “relative” variant of both Buchberger’s criteria for avoiding reductions to zero and Schreyer’s theorem for a Grobner basis of the syzygy module. As main contribution, we then introduce the novel notion of relative involutive bases and present an algorithm for their explicit construction....