A new algorithm for Gröbner bases conversion

Abstract

A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering.

The main ingredient of the new algorithm is the computation of a Gröbner basis with respect to the target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the ideal with respect to the target ordering. In general, more than one iteration may be needed to get a Gröbner basis with respect to the target ordering since truncated polynomials may miss some leading monomials.

The new algorithm has been implemented in Maple and its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches in Maple. In practice, a Gröbner basis with respect to a target ordering can be computed in a single iteration on most examples.

Since the proposed basis conversion algorithm uses simple concepts of Gröbner basis theory, it can be easily taught in contrast to methods based on Gröbner walk.