Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
James Miller · Cambridge University Press eBooks · 1998
Gröbner bases have been an important tool for performing computations in polynomial rings since Buchberger first presented an algorithm to construct them in 1965. A generalized type of Gröbner basis was developed in the context of a valuation ring by Sweedler in 1988. The purpose of this paper is to concretize Sweedler's theory to the context of a k -subalgebra of a polynomial ring k [ x 1 ,…, x n ], where k is a field and to present effective additional algorithms necessary for intrinsically constructing objects that we shall call SAGBI-Gröbner bases. Introduction SAGBI-Gröbner bases are simply the counterparts in ideals of subalgebras of polynomial rings to Gröbner bases of ideals of the full polynomial ring. Our intrinsic development of SAGBI-Gröbner theory will mimic that of ordinary Gröbner bases, an approach that has a certain aesthetic appeal. In addition, this parallel development provides another point of view from which to study the mechanics of Gröbner basis theory, which was first presented in an algorithmic fashion by Buchberger (1965) (see also Buchberger (1985)). We assume little experience with Gröbner bases on the part of the reader; an overview of the subject may be found in the texts of Adams and Loustaunau (1994), Becker and Weispfennig (1993), and Cox, Little, and O'Shea (1992).