On a family of Laurent polynomials generated by 2x2 matrices

TL;DR

This paper studies a family of Laurent polynomials generated by 2x2 matrices, expressing them in terms of parameters and Chebyshev polynomials, and describing their zero sets.

Contribution

It introduces a natural parametrization of the Laurent polynomial family generated by 2x2 matrices and relates these polynomials to Chebyshev polynomials and their zero sets.

Findings

The family of Laurent polynomials is three-parametric.

Explicit formulas relate polynomials to Chebyshev polynomials.

The zero set of these polynomials is characterized.

Abstract

To a $2\times2$ matrix $G$ with complex entries, we relate the sequence of Laurent polynomial $L_n(z,G)=\tr \big(G\big[\begin{smallmatrix}z&0\\ 0&z^{-1}\end{smallmatrix}\big]G^{\ast}\big)^n$. It turns out that for each \(n\), the family $\big\{L_n(z,G)\big\}_G$, where $G$ runs over the set of all $2\times2$ matrices, is a three-parametric family. A natural parametrization of this family is found. The polynomial $L_n(z,G)$ is expressed in terms of these parameters and the Chebyshev polynomial $T_n$. The zero set of the polynomial $L_n(z,G)$ is described.