  Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate
polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary
parts of its roots are not contained in the relative interior of $K$. If $K$ is
the non-negative orthant, $K$-stability specializes to the usual notion of
stability of polynomials.
  We develop generalizations of preservation operations and of combinatorial
criteria from usual stability towards conic stability. A particular focus is on
the cone of positive semidefinite matrices (psd-stability). In particular, we
prove the preservation of psd-stability under a natural generalization of the
inversion operator. Moreover, we give conditions on the support of psd-stable
polynomials and characterize the support of special families of psd-stable
polynomials.
