2019/08/29 by Dey, Papri, Gardoll, Stephan, Theobald, Thorsten · 1 citation
#12D10 #14P10 (Primary) 26C10 #30C15 #90C22 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.11124
Given a proper cone K ⊆ ℝn, a multivariate polynomial f ∈ ℂ[z] = ℂ[z1, …, zn] 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.