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. The composition of Schur-Szegö of the polynomials P = n j=0 C j n ajx j andIn 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).

A mapping connected with the Schur–Szegő composition

“…(see [1]) where a j ∈ C are unique up to permutation. If P is real, some of the numbers a j are real, the rest form conjugate couples (otherwise conjugation of both sides of (1) defines a new set of numbers a j ).…”

“…La CSS est commutative et associative (voir [5] pour plus de détails sur la CSS). Dans l'article [1] on prouve que chaque polynôme P de degré n et tel que P (−1) = 0, est présentable sous la forme P = K a 1 * n · · · * n K a n−1 (K) Remarque 1. (1) Posons b i := −i/(n − i).…”

A realization theorem in the context of the Schur-Szegő composition

“…, σ n−1 be the elementary symmetric polynomials in the numbers a j defined in (2). It follows from the proof of Proposition 3 in [3] that the mapping (σ 1 , . .…”

“…One can readily show that these numbers continuously depend on the coefficients of P . Note that if the polynomial K a is represented in the form n j=0 n j θ j x j , then the numbers θ j form an arithmetic progression; see (3). In the present note, we consider the case in which P is a real polynomial.…”

“…(If zero is not an n-fold root of P , then one can always ensure that P (−1) = 0 by making a linear homogeneous change of the variable x followed by a multiplication of P by a nonzero constant to keep it monic.) It was shown in [3] (see also [4]) that P can be represented in the form P = K a 1 * · · · * K a n−1 , (2) where K a = (x + 1) n−1 (x + a) and the numbers a j are uniquely determined up to permutation. The same paper provides examples showing that the multiplicities of these numbers and their being real are not directly related to the respective properties of the roots of P .…”