Operator Bezoutiant and Roots of Entire Functions

Abstract

To every two polynomials f (z) and g(z) of order n, Sylvester assigns the square formula

$$\mathfrak{B}(f,g,{x_0}, \ldots ,{x_{n - 1}}) = \sum\limits_{k,l = 1}^n {{c_{kl}}{x_k}{x_l}}$$

(0.1)

called by him the Bezoutiant. Here the equality

$$\frac{{f(x)g(y) - f(y)g(x)}}{{x - y}} = \sum\limits_{k,l = 0}^{n - 1} {{c_{kl}}{x_k}{x_l}}$$

(0.2)

is true [46]. The Bezoutiant ℬ is used in order to define the number of common roots of f (z) and g(z). The same form ℬ is used when deducing the Schur-Cohn theorem in which the distribution of roots of the polynomial f (z) with respect to the circle |z| = 1 is clarified. Krein extended the Schur-Cohn theorem to entire functions of the form

$$F(z) = 1 + \int\limits_0^\omega {{e^{izt}}\overline {\Phi (t)} dt,} \Phi (t) \in L(0,\omega )$$

(0.3)