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arXiv · 2112.13321

Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

Grigoriy Blekherman·Mario Kummer·Raman Sanyal·Kevin Shu·Shengding Sun
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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.

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Grigoriy Blekherman, Mario Kummer, Raman Sanyal, Kevin Shu, Shengding Sun. 2021-12-26. Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment. https://doi.org/10.1093/imrn%2Frnac291

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