Nuij type pencils of hyperbolic polynomials

Abstract

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+s{p}^{\prime }$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s):=p(z)+{\sum }_{k=1}^d{a}_k{s}^k{p}^{(k)}(z)$. 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.

References in corpus (2)

• The Lax conjecture is true
• Polya-Schur master theorems for circular domains and their boundaries