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Conic stability of polynomials and positive maps

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 $K$.