二元稳定实零多项式
Stable and real-zero polynomials in two variables
摘要：
对于二次$(n_1, n_2)$的二元多项式$p(z_1, z_2)$，其中$p(0,0)=1$在开单位双圆盘中没有零，我们构造了一个形式为$$p(z_1,z_2)=\det (I - K Z),$$的行列式表示，其中$Z$是一个$(n_1+n_2)\times(n_1+n_2)$对角矩阵，其坐标变量$z_1$，$z_2$在对角线上，$K$是一个压缩。我们表明，$K$可以被选择为酉当且仅当$p$是一个（么模）常数倍，其反向。此外，对于每一个具有$p(0,0)=1$的二元实零多项式$p(x_1, x_2),$，我们提供了一个构造来构建形式为$$p(x_1,x_2)=\det (I+x_1A_1+x_2A_2),$$的表示，其中$A_1$和$A_2$是大小等于$p$的次数的厄米特矩阵.这两种构造的一个关键组成部分是一个稳定的因子分解的半正定矩阵值多项式在一个变量，无论是在圆（三角多项式）或在实数线（代数多项式）。
英文摘要：
For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an $(n_1+n_2)\times(n_1+n_2)$ diagonal matrix with coordinate variables $z_1$, $z_2$ on the diagonal and $K$ is a contraction. We show that $K$ may be chosen to be unitary if and only if $p$ is a (unimodular) constant multiple of its reverse. Furthermore, for every bivariate real-zero polynomial $p(x_1, x_2),$ with $p(0,0)=1$, we provide a construction to build a representation of the form $$p(x_1,x_2)=\det (I+x_1A_1+x_2A_2),$$ where $A_1$ and $A_2$ are Hermitian matrices of size equal to the degree of $p$. A key component of both constructions is a stable factorization of a positive semidefinite matrix-valued polynomial in one variable, either on the circle (trigonometric polynomial) or on the real line (algebraic polynomial).
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