Several Applications of Bezout Matrices

Abstract

The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting nonselfadjoint operators, boundaries of quadrature domains etc. We present a survey of several properties of Bezout matrices and their applications in all mentioned topics. We use the framework of Vandermonde vectors because such approach allows us to give new proofs of both classical and modern results and in many cases to obtain new explicit formulas. These explicit formulas can significantly simplify various computational problems and, in particular, make the research of algebraic curves and their applications easier. In addition we wrote a Maple software package, which computes all the formulas. For instance, as Bezout matrices are used in order to compute the image of a rational transformation of an algebraic curve, we used these results to study some connections between small degree rational transformation of an algebraic curve and the braid monodromy of its image.