On the logarithmic energy of solutions to the polynomial eigenvalue problem
Probability
2025-05-19 v4 Numerical Analysis
Numerical Analysis
Abstract
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.
Cite
@article{arxiv.2408.11148,
title = {On the logarithmic energy of solutions to the polynomial eigenvalue problem},
author = {Diego Armentano and Federico Carrasco and Marcelo Fiori},
journal= {arXiv preprint arXiv:2408.11148},
year = {2025}
}