矩阵多球上稳定多项式的压缩行列式表示
Contractive determinantal representations of stable polynomials on a matrix polyball
摘要：
我们证明了一个不可约多项式$p$没有零的封闭矩阵单位polyball，a.k.a.类型I的Cartan域的Carnival积，并且使得$p(0)=1$，允许严格压缩行列式表示，即，其中$p=\det(I-KZ_n)$是非负整数的$k$元组，$Z_n=\bigoplus_{r=1}^k(Z^{(r)}\otimes I_{n_r})$、$Z^{(r)}=[z^{(r)}_{ij}]$是复矩阵，$p$是矩阵项$z^{(r)}_{ij}$中的多项式，并且$K$是严格压缩矩阵。这个结果是通过非交换提升和极小非交换结构系统实现的奇点定理得到的。
英文摘要：
We show that an irreducible polynomial $p$ with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that $p(0)=1$, admits a strictly contractive determinantal representation, i.e., $p=\det(I-KZ_n)$, where $n=(n_1,...,n_k)$ is a $k$-tuple of nonnegative integers, $Z_n=\bigoplus_{r=1}^k(Z^{(r)}\otimes I_{n_r})$, $Z^{(r)}=[z^{(r)}_{ij}]$ are complex matrices, $p$ is a polynomial in the matrix entries $z^{(r)}_{ij}$, and $K$ is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.
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