Universal Gröbner Basis for Parametric Polynomial Ideals

Abstract

In this paper, we introduce the concept of universal Gröbner basis for a parametric polynomial ideal. In this direction, we present a new algorithm, called UGS, which takes as input a finite set of parametric polynomials and outputs a universal Gröbner system for the ideal generated by input polynomials, by decomposing the space of parameters into a finite set of parametric cells and for each cell associating a finite set of parametric polynomials which is a universal Gröbner basis for the ideal corresponding to that cell. Indeed, for each values of parameters satisfying a condition set, the corresponding polynomial set forms a universal Gröbner basis for the ideal. Our method relies on the parametric variant of the Gröbner basis conversion and also on the PGBMain algorithm due to Kapur et al. to compute parametric Gröbner bases. The proposed UGS algorithm has been implemented in Maple-Sage and its performance is investigated through an example.