ar X iv : m at h / 06 01 04 7 v 1 [ m at h . A G ] 3 J an 2 00 6 SEVERAL APPLICATIONS OF BEZOUT MATRICES

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. Introduction Numerous works in operator theory shows that an algebraic curve given by a determinantal representation can be associated to a system of commuting nonselfadjoint operators. In the simplest case there are two commuting nonselfadjoint operators in a system and they are rational images of the same operator. This particular case gives a motivation and a framework for the studying the image of a complex line under a rational transformation. Using the notion of a determinantal representation of an algebraic curve there was found the explicit formula which describes the image of a complex line under a rational transformation. The image is given by a determinantal representation which uses the Bezout matrices of pairs of polynomials that define the rational transformation. This property of Bezout matrices can be used for many purposes, i.e. to study the braid monodromy of the image of two intersecting lines under rational transformations of small degrees. Also using this property one can describe explicitly the boundary of a quadrature domain. This property and many others can be proved using the Vandermonde vectors, which are a natural framework to study Bezout matrices. These properties were used by the authors of this paper to create a package of procedures for Maple software [21]. This package allows the computation of all Partially supported by EU-network HPRN-CT-2009-00099(EAGER), (The Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany), the Israel Science Foundation grant # 8008/02-3 (Excellency Center ”Group Theoretic Methods in the Study of Algebraic Varieties”). The authors wish to thank Prof. Victor Vinnikov for helpful advices. 1 2 SHMUEL KAPLAN, ALEXANDER SHAPIRO AND MINA TEICHER classical and modern formulas that are mentioned in the paper, and to perform many different tasks. 1. Bezout matrices and Vandermonde vectors First example of Bezout matrices for polynomials of small degree appeared in Euler’s work in 1748, [7]. Using this example Bezout gave a general definition of Bezout matrices for polynomials of any degree in 1764, [2]. The notation of Bezoutian matrix was introduced by Sylvester in 1853, [23]. The most common definition was given by Cayley in 1857, [4]. Definition 1.1. For two polynomials in one variable p(x) and q(x) of degree n there exists uniquely determined n × n symmetric matrix B(p, q) = (bij) n i,j=1 such that p(x)q(y)− q(x)p(y) x− y = n