Conic stability of polynomials and positive maps

[摘要] Given a proper cone K subset of R-n, a multivariate polynomial f epsilon C[z] = C[z(1), . . . , 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. (C) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2021-07-01

[效力级别]  Proceedings Paper