  Given a proper cone $K \subseteq \mathbb{R}^n$, a multivariate polynomial $f
\in \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_n]$ is called $K$-stable if it
does not have a root whose vector of the imaginary parts is contained in the
interior of $K$. If $K$ is the non-negative orthant, then $K$-stability
specializes to the usual notion of stability of polynomials.
  We study conditions and certificates for the $K$-stability of a given
polynomial $f$, especially for the case of determinantal polynomials as well as
for quadratic polynomials. A particular focus is on psd-stability. For cones
$K$ with a spectrahedral representation, we construct a semidefinite
feasibility problem, which, in the case of feasibility, certifies $K$-stability
of $f$. This reduction to a semidefinite problem builds upon techniques from
the connection of containment of spectrahedra and positive maps.
  In the case of psd-stability, if the criterion is satisfied, we can
explicitly construct a determinantal representation of the given polynomial. We
also show that under certain conditions, for a $K$-stable polynomial $f$, the
criterion is at least fulfilled for some scaled version of $K$.
