Schur--Horn type inequalities for hyperbolic polynomials
Teng Zhang
TL;DR
The paper develops a hyperbolic analogue of the Schur--Horn principle for symmetric hyperbolic polynomials $P$ by proving that $P^{1/k}$ is concave and Schur-concave on the hyperbolicity cone $\Gamma(P)$, enabling majorization-based comparisons between diagonals and spectra. This leads to a resolution of Nam Le's Hadamard-type conjecture: if $\lambda(A)\in\Gamma(P)$ then $\mathrm{diag}(A)\in\Gamma(P)$ and $P(\mathrm{diag}(A))\ge P(\lambda(A))$, with the inequality sharpening classical eigenvalue bounds. The authors further introduce a Reynolds-group symmetrization principle on hyperbolicity cones, showing that averaging over a finite group action preserves the cone and strictly increases the hyperbolic polynomial under concavity; this yields concise proofs of hyperbolic Fischer--Hadamard inequalities for PSD-stable linear principal minor polynomials. Together, these results provide a unified framework that lifts eigenvalue majorization results to general symmetric hyperbolic polynomials and highlight the central role of symmetry and concavity in hyperbolic positivity and inequality theory.
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 Gårding'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.