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arXiv · 2206.10913

Combinatorics and preservation of conically stable polynomials

Giulia Codenotti·Stephan Gardoll·Thorsten Theobald
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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.

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Giulia Codenotti, Stephan Gardoll, Thorsten Theobald. 2022-06-22. Combinatorics and preservation of conically stable polynomials. https://arxiv.org/abs/2206.10913

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