An extended GCD algorithm for parametric univariate polynomials and application to parametric smith normal form

Abstract

An extended greatest common divisor (GCD) algorithm for parametric univariate polynomials is presented in this paper. This algorithm computes not only the GCD of parametric univariate polynomials in each constructible set but also the corresponding representation coefficients (or multipliers) for the GCD expressed as a linear combination of these parametric univariate polynomials. The key idea of our algorithm is that for non-parametric case the GCD of arbitrary finite number of univariate polynomials can be obtained by computing the minimal Gröbner basis of the ideal generated by those polynomials. But instead of computing the Gröbner basis of the ideal generated by those polynomials directly, we construct a special module by adding the unit vectors which can record the representation coefficients, then obtain the GCD and representation coefficients by computing a Gröbner basis of the module. This method can be naturally generalized to the parametric case because of the comprehensive Gröbner systems for modules. As a consequence, we obtain an extended GCD algorithm for parametric univariate polynomials. More importantly, we apply the proposed extended GCD algorithm to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form.