Properties of conically stable polynomials and imaginary projections

Abstract

We thoroughly study the properties of conically stable polynomials and imaginary projections. A multivariate complex polynomial is called stable if its nonzero whenever all coordinates of the respective argument have a positive imaginary part. In this dissertation we consider the generalized notion of K-stability. A multivariate complex polynomial is called K-stable if its non-zero whenever the imaginary part of the respective argument lies in the relative interior of the cone K. We study connections to various other objects, including imaginary projections as well as preservers and combinatorial criteria for conically stable polynomials.