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
Main Authors Armentano, Diego, Carrasco, Federico, Fiori, Marcelo
Format Journal Article
LanguageEnglish
Published 20.08.2024
Subjects
Computer Science - Numerical Analysis
Mathematics - Numerical Analysis
Mathematics - Probability
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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.
DOI:10.48550/arxiv.2408.11148