Schur--Horn type inequalities for hyperbolic polynomials
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914257716641792 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Schur--Horn type inequalities for hyperbolic polynomials Zhang, Teng Functional Analysis Representation Theory 15A42 (Primary), 26C10, 13A50 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. |
| title | Schur--Horn type inequalities for hyperbolic polynomials |
| topic | Functional Analysis Representation Theory 15A42 (Primary), 26C10, 13A50 |
| url | https://arxiv.org/abs/2601.10602 |