Combinatorics and preservation of conically stable polynomials

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