A hybrid approach to the computation of the inertia of a parametric family of Bezoutians with application to some stability problems for bivariate polynomials Original Research Article

Given two polynomials with coefficients over image[k], the associated Bezout matrix B(k) with entries over image[k] defines a parametric family of Bezout matrices with entries over image. It is intended in this paper to propose a hybrid approach for determining the inertia of B(k) for any value of k in some real interval. This yields an efficient solution to certain root-location problems for bivariate polynomials. We first develop a fast fraction-free method for computing an inverse triangular factorization of the Bezout matrix B(k) over the integral domain image[k]. In this way, we may easily compute the sequence {φi(k)} of the training principal minors of B(k). For almost any value image of k the associated sign sequence image specifies the inertia of image. The function sign (φi(k)) is final obtained by numerically computing rational approximations of the real zeros of φi(k) ε image[k].