Recent advances on determining the number of real roots of parametric polynomials

An explicit criterion for the determination of the numbers and multiplicities of the real/imaginary roots for polynomials with symbolic coefficients is based on a Complete Discrimination System (CDS). A CDS is a set of explicit expressions in terms of the coefficients that are sufficient for determining the numbers and multiplicities of the real and imaginary roots. Basically, the problem is considered on a total real axis and a total complex plane. However, it is often required in both practice and theory to determine the number of real roots in some interval, especially (0, infinity) or (-infinity,0). This article is mainly devoted to solving the case in an interval, but some global results are reviewed for understanding. It is shown, with examples, how useful the CDS can be in order to understand the behaviour of the roots of an univariate polynomial in terms of the coefficients. (C) 1999 Academic Press.