A Hybrid Approach to the Computation of the Inertia of a Parametric Family of Bezoutians with Application to Some Stability Problems for Bivariate Polynomials

Given two polynomials with coeecients over Zk], the associated Bezout matrix B(k) with entries over Zk] deenes a parametric family of Bezout matrices with entries over Z. 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 eecient solution to certain root-location problems for bivariate polynomials. We rst develop a fast fraction-free method for computing an inverse triangular factorization of the Bezout matrix B(k) over the integral domain Zk]. In this way, we may easily compute the sequence fi(k)g of the trailing principal minors of B(k). For almost any value k of k the associated sign sequence fsign(i(k))g speciies the inertia of B(k). The function sign(i(k)) is nally obtained by numerically computing rational approximations of the real zeros of i(k) 2 Zk].