Conic stability of polynomials

TL;DR

This paper introduces the concept of conic stability for multivariate polynomials, generalizing existing stability notions, and provides new characterizations and criteria related to this broader stability framework.

Contribution

It generalizes the Hermite-Kakeya-Obreschkoff Theorem and stability criteria for determinantal polynomials to the conic stability setting.

Findings

Generalized multivariate stability concepts to conic stability.

Provided a characterization via directional Wronskian.

Extended stability criteria to positive semidefinite cone.

Abstract

We introduce and study the notion of conic stability of multivariate complex polynomials in $\mathbb{C}[z_1,\ldots, z_n]$, which naturally generalizes the stability of multivariate polynomials. In particular, we generalize Borcea's and Br\"and\'en's multivariate version of the Hermite-Kakeya-Obreschkoff Theorem to the conic stability and provide a characterization in terms of a directional Wronskian. And we generalize a major criterion for stability of determinantal polynomials to stability with respect to the positive semidefinite cone.