An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD

Abstract

A new efficient algorithm for computing a parametric greatest common divisor (GCD) of parametric multivariate polynomials over k[u][x] is presented. The algorithm is based on a well-known simple insight that the GCD of two multivariate polynomials (non-parametric as well as parametric) can be extracted using the generator of the quotient ideal of a polynomial with respect to the second polynomial. And, further, this generator can be obtained by computing a minimal Gröbner basis of the quotient ideal. The main attraction of this idea is that it generalizes to the parametric case for which a comprehensive Gröbner basis is constructed for the parametric quotient ideal. It is proved that in a minimal comprehensive Gröbner system of a parametric quotient ideal, each branch of specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric GCD of that branch is obtained by division. This algorithm does not need to consider whether parametric polynomials are primitive w.r.t. the main variable. This is in sharp contrast to two algorithms recently proposed by Nagasaka (ISSAC, 2017). The resulting algorithm is not only conceptually simple to understand but is considerably efficient. The proposed algorithm and both of Nagasaka's algorithms have been implemented in Singular (available at http://www.mmrc.iss.ac.cn/~dwang/software.html), and their performance is compared on a number of examples. For more than two polynomials, this process can be repeated by considering pairs of polynomials; the efficiency in that case becomes even more evident.