Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866914876621848576 |
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| author | Blekherman, Grigoriy Kummer, Mario Sanyal, Raman Shu, Kevin Sun, Shengding |
| author_facet | Blekherman, Grigoriy Kummer, Mario Sanyal, Raman Shu, Kevin Sun, Shengding |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_13321 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment Blekherman, Grigoriy Kummer, Mario Sanyal, Raman Shu, Kevin Sun, Shengding Algebraic Geometry Rings and Algebras 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. |
| title | Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment |
| topic | Algebraic Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2112.13321 |