Full paper in PDF format:
$%S. Alkhatib and V. P. Kostov, The Schur-Szegö composition of real polynomials of degree 2, Rev. Mat. Complut. 21 (2008), no. 1, 191–206.%$

The Schur-Szegö Composition of Real Polynomials of Degree 2
Soliman ALKHATIB and Vladimir Petrov KOSTOV
Université de Nice
Laboratoire de Mathématiques
Parc Valrose
06108 Nice Cedex 2 — France

khatib@math.unice.fr  kostov@math.unice.fr

Received: April 16, 2007
Accepted: September 17, 2007

ABSTRACT

A real polynomial P  in one real variable is hyperbolic if its roots are all real. The composition of Schur-Szegö of the polynomials    ∑n    j   j
P =  j=0Cnajx  and     ∑n    j  j
Q =   j=0C nbjx  is the polynomial        ∑n    j    j
P *Q =   j=0Cnajbjx  . 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)  .

Key words: composition of Schur-Szegö, hyperbolic polynomial.
2000 Mathematics Subject Classification:
12D10.