Combinatorics and preservation of conically stable polynomials

Given a closed, convex cone K ⊆ R n , a multivariate polynomial f ∈ C [ 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 nonnegative 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 toward 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.