Determinantal representations of invariant hyperbolic plane curves

Abstract

We study hyperbolic polynomials with nice symmetry and express them as the determinant of a Hermitian matrix with special structure. The goal of this paper is to answer a question posed by Chien and Nakazato in 2015. By properly modifying a determinantal representation construction of Dixon (1902), we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix. Moreover, if the polynomial is invariant under the action of the dihedral group of order n, the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries.