关于由2 × 2矩阵生成的一类Laurent多项式
On a family of Laurent polynomials generated by 2x2 matrices
摘要：
对于具有复数元素的$2\times2$矩阵$G$，我们将Laurent多项式$L_n(z,G)=\tr \big(G\big[\begin{smallmatrix}z&0\\ 0&z^{-1}\end{smallmatrix}\big]G^{\ast}\big)^n$的序列联系起来。事实证明，对于每个\(n\)，族$\big\{L_n(z,G)\big\}_G$，其中$G$在所有$2\times2$矩阵的集合上运行，是一个三参数族。找到了该族的一个自然的参数化形式，多项式$L_n(z,G)$用这些参数和切比雪夫多项式$T_n$表示.描述了多项式$L_n(z,G)$的零点集。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
