» Yang L. - Determining the number of real roots of parametric polynomials
Yang L. - Determining the number of real roots of parametric polynomials
||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, oo) or (—00, 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.
© 1999 Academic Press
It is of theoretical and practical significance to determine the number of roots in a certain range of a polynomial by an explicit criterion. The classical Sturm Theorem is an on-line algorithm efficient for determining the numbers of real roots of constant coefficient polynomials, but very inconvenient for those with symbolic/literal coefficients. This shortcoming prevents computer implementations of Tarski's and other methods in automated theorem proving wherein Sturm's algorithm is employed. After all, Sturm's algorithm is not an explicit criterion based on a complete discrimination system.
One knows that a discriminant of a polynomial is an explicit expression in terms of the coefficients, which is used to decide whether the given polynomial has repeated roots or not. For example, the discriminant of ax2 + bx + с is b2 — Aac. However, a single discriminant could not give more detailed information about the roots of a polynomial with a higher degree. A Complete Discrimination System (CDS) is a set of explicit expressions in terms of the coefficients, which is sufficient for determining the numbers and multiplicities of the real and imaginary roots, namely, determining the complete root classification. One can find from some articles, e.g. Arnon (1988), a complete root classification based on a CDS for a quartic polynomial with symbolic coefficients. Arnon used that result to derive the conditions on real numbers p, q, r such that