Interlacing properties and the Schur-Szeg\H{o} composition

TL;DR

This paper explores the interlacing properties of roots of hyperbolic polynomials related to the Schur-Szegő composition, revealing positive rational eigenvalues and eigenvectors defined by hyperbolic polynomials.

Contribution

It establishes interlacing properties of roots in hyperbolic polynomials associated with the Schur-Szegő composition, expanding understanding of their spectral and root structure.

Findings

Eigenvalues of the affine mapping are positive rational numbers.

Eigenvectors are characterized by hyperbolic polynomials with real roots.

Interlacing properties of roots are proven for these polynomials.

Abstract

Each degree $n$ polynomial in one variable of the form $(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable in a unique way as a Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, see \cite{Ko1}, \cite{AlKo} and \cite{Ko2}. Set $\sigma _j:=\sum _{1\leq i_1<\cdots <i_j\leq n-1}a_{i_1}\cdots a_{i_j}$. The eigenvalues of the affine mapping $(c_1,\ldots ,c_{n-1})\mapsto (\sigma _1,\ldots ,\sigma _{n-1})$ are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.