Invariant hyperbolic curves: determinantal representations and applications to the numerical range

TL;DR

This paper explores invariant hyperbolic plane curves, demonstrating they possess determinantal representations with applications to understanding matrices with rotationally invariant numerical ranges.

Contribution

It establishes that all invariant hyperbolic plane curves have determinantal representations via block cyclic weighted shift matrices, extending prior results.

Findings

Invariant hyperbolic curves have determinantal representations.

Numerical range invariance implies representation by block cyclic weighted shift matrices.

Analogue of Nuij's theorem for invariant hyperbolic polynomials.

Abstract

Here we study the space of real hyperbolic plane curves that are invariant under actions of the cyclic and dihedral groups and show they have determinantal representations that certify this invariance. We show an analogue of Nuij's theorem for the set of invariant hyperbolic polynomials of a given degree. The main theorem is that every invariant hyperbolic plane curve has a determinantal representation using a block cyclic weighted shift matrix. This generalizes previous work by Lentzos and the first author, as well as by Chien and Nakazato. One consequence is that if the numerical range of a matrix is invariant under rotation, then it is the numerical range of a block cyclic weighted shift matrix.