Determinantal representations of invariant hyperbolic plane curves

TL;DR

This paper demonstrates that hyperbolic polynomials with certain symmetries can be represented as determinants of structured Hermitian matrices, extending classical results and answering a question from 2015.

Contribution

It provides explicit determinantal representations for symmetric hyperbolic polynomials using cyclic weighted shift matrices, generalizing Dixon's construction.

Findings

Hyperbolic polynomials invariant under cyclic groups have determinantal representations with cyclic weighted shift matrices.

For dihedral group invariance, the matrices can be chosen to have real entries.

The results extend classical determinantal representation theory to symmetric hyperbolic polynomials.

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.