On the logarithmic energy of solutions to the polynomial eigenvalue problem

TL;DR

This paper analyzes the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices, generalizing known results and establishing bounds between extremal cases.

Contribution

It introduces a general framework for computing logarithmic energy in polynomial eigenvalue problems, extending previous specific cases to a broader class.

Findings

Logarithmic energy of solutions is bounded between Shub-Smale and spherical ensemble cases.

Roots of Shub-Smale polynomials have the lowest logarithmic energy.

Generalization of known extremal cases to broader polynomial eigenvalue problems.

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.