  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).
