Stable and real-zero polynomials in two variables

Abstract

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 $$p(z_1,z_2)=\det (I-KZ),$$ 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 $$p(x_1,x_2)=\det (I+x_1{A}_1+x_2{A}_2),$$ 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).

References in corpus (1)

• Determinantal representations of hyperbolic plane curves: An elementary approach