Schur--Horn type inequalities for hyperbolic polynomials

TL;DR

This paper proves a Schur--Horn type inequality for symmetric hyperbolic polynomials, resolving a conjecture on Hadamard inequalities and developing a symmetrization principle for hyperbolicity cones, with applications to Fischer--Hadamard inequalities.

Contribution

It introduces a new Schur--Horn type inequality for hyperbolic polynomials and a symmetrization principle on hyperbolicity cones, extending classical eigenvalue inequalities.

Findings

Resolved a conjecture of Nam Q. Le on Hadamard inequalities for hyperbolic polynomials.

Developed a symmetrization principle for hyperbolicity cones under group actions.

Provided a short proof of hyperbolic Fischer--Hadamard inequalities for PSD-stable linear principal minor polynomials.

Abstract

We establish a Schur--Horn type inequality for symmetric hyperbolic polynomials. As an immediate consequence, we resolve a conjecture of Nam Q. Le on Hadamard-type inequalities for hyperbolic polynomials. Our argument is based on the Schur--Horn theorem, the Birkhoff theorem, and G{\aa}rding's concavity theorem for hyperbolicity cones. Beyond the eigenvalue level, we develop a symmetrization principle on hyperbolicity cones: if a hyperbolic polynomial is invariant under a finite group action, then its value increases under the associated Reynolds operator (group averaging). Applied to the sign-flip symmetries of linear principal minor polynomials introduced by Blekherman et al., this yields a short proof of the hyperbolic Fischer--Hadamard inequalities for PSD-stable lpm polynomials.