Conic stability of polynomials and positive maps

Given a proper cone K?Rn, multivariate polynomial f?C[z]=C[z1,…,zn] is called K-stable if it does not have root whose vector of the imaginary parts contained in interior K. If K non-negative orthant, then K-stability specializes to usual notion stability polynomials. We study conditions and certificates for given f, especially case determinantal polynomials as well quadratic A particular focus on psd-stability. For cones with spectrahedral representation, we construct semidefinite feasibility problem, which, feasibility, certifies f. This reduction problem builds upon techniques from connection containment spectrahedra positive maps. In psd-stability, criterion satisfied, can explicitly representation polynomial. also show that under certain conditions, at least fulfilled some scaled version