Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

TL;DR

This paper introduces linear principal minor polynomials (lpm) as a new class of polynomials linked to symmetric matrices, establishing their connection to stable polynomials, and generalizing classical inequalities to this broader setting.

Contribution

It establishes a bijection between multiaffine stable polynomials and PSD-stable lpm polynomials, enabling new hyperbolic polynomial constructions and generalizations of classical inequalities.

Findings

Established a one-to-one correspondence between stable polynomials and lpm polynomials.

Generalized Fisher--Hadamard and Koteljanskii inequalities to PSD-stable lpm polynomials.

Proposed the spectral containment conjecture relating eigenvalues and lpm polynomial evaluations.

Abstract

A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection to multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to generalize the well-known Fisher--Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.