On the logarithmic energy of solutions to the polynomial eigenvalue problem

In this paper, we compute the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices. We generalize some known results for the Shub-Smale polynomials, and the spherical ensemble. These two processes are the two extremal particular cases of the polynomial ei...

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Bibliographic Details
Published inarXiv.org
Main Authors Armentano, Diego, Carrasco, Federico, Fiori, Marcelo
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 22.08.2024
Subjects
Eigenvalues
Logarithms
Polynomials
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Summary:In this paper, we compute the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices. We generalize some known results for the Shub-Smale polynomials, and the spherical ensemble. These two processes are the two extremal particular cases of the polynomial eigenvalue problem, and we prove that the logarithmic energy lies between these two cases. In particular, the roots of the Shub-Smale polynomials are the ones with the lowest logarithmic energy of the family.
ISSN:2331-8422