Computing Independent Variable Sets for Polynomial Ideals

Abstract

Computing independent variable sets for polynomial ideals plays an important role in solving high-dimensional polynomial equations. The computation of a Gröbner basis for an ideal, with respect to a block lexicographical order in classic methods, is huge, and then an improved algorithm is given. Based on the quasi-Gröbner basis of the extended ideal, a criterion of assigning independent variables is gained. According to the criteria, a maximal independent variable set for a polynomial ideal can be computed by assigning indeterminates gradually. The key point of the algorithm is to reduce dimensions so that the unit of computation is one variable instead of a set, which turns a multivariate problem into a single-variable problem and turns the computation of rational function field into that of the fundamental number field. Hence, the computation complexity is reduced. The algorithm has been analysed by an example, and the results reveal that the algorithm is correct and effective.