Combinatorics and preservation of conically stable polynomials

TL;DR

This paper extends the concept of polynomial stability to conic stability, especially focusing on positive semidefinite matrices, and explores preservation operations, combinatorial criteria, and support characterizations for these polynomials.

Contribution

It introduces generalizations of stability-preserving operations and criteria from classical stability to conic stability, with a focus on psd-stability and support characterization.

Findings

Psd-stability is preserved under a generalized inversion operator.

Conditions for the support of psd-stable polynomials are established.

Characterization of support for specific psd-stable polynomial families.

Abstract

Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the non-negative orthant, $K$-stability specializes to the usual notion of stability of polynomials. We develop generalizations of preservation operations and of combinatorial criteria from usual stability towards conic stability. A particular focus is on the cone of positive semidefinite matrices (psd-stability). In particular, we prove the preservation of psd-stability under a natural generalization of the inversion operator. Moreover, we give conditions on the support of psd-stable polynomials and characterize the support of special families of psd-stable polynomials.