RMC - Vol. 21 Num. 1 - S. Alkhatib/V. P. Kostov

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.