The Schur-Szegö Composition of Real Polynomials of Degree 2

Abstract

A real polynomial P in one real variable is hyperbolic if its roots are all real. n j j The composition of Schur-Szeg¨ of the polynomials P = o j =0 Cn aj x and n n j j j j Q = j =0 Cn bj x is the polynomial P Q = j =0 Cn aj bj x . In the present paper we show how for n = 2 and when P and Q are real or hyperbolic the roots of P Q depend on the roots or the coefficients of P and Q. We consider also the case when n 2 is arbitrary and P and Q are of the form (x - 1)n-1 (x + b). This case is interesting in the context of the possibility to present every polynomial having one of its roots at (-1) as a composition of n - 1 polynomials of the form (x + 1)n-1 (x + b).