A note on Computing SAGBI-Grobner bases in a Polynomial Ring over a Field

Abstract

In the paper [2] Miller has made concrete Sweedler's theory for ideal bases in commutative valuation rings (see [5]) to the case of subalgebras of a polynomial ring over a field, the ideal bases are called SAGBI-Grobner bases in this case. Miller proves a concrete algorithm to construct and verify a SAGBI-Grobner basis, given a set of generators for an ideal in the subalgebra. The purpose of this note is to present an observation which justifies substantial shrinking of the so called syzygy family of a pair of polynomials. Fewer elements in the syzygy family means that fewer syzygy-polynomials need to be checked in the SAGBI-Grobner basis construction/verification algorithm, thus decreasing the time needed for computation.