Nuij type pencils of hyperbolic polynomials

TL;DR

This paper characterizes specific perturbations of hyperbolic polynomials that preserve hyperbolicity and admits universal determinantal representations, revealing connections to symmetric Toeplitz matrices.

Contribution

It provides a complete characterization of perturbations maintaining hyperbolicity and the existence of universal determinantal representations for these polynomial families.

Findings

Characterization of perturbations preserving hyperbolicity.

Identification of conditions for universal determinantal representations.

Connection to symmetric Toeplitz matrices.

Abstract

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices.