Expected energy of zeros of elliptic polynomials

TL;DR

This paper extends the analysis of zeros of elliptic polynomials to compute expected Riesz energies and separation distances, providing new insights into their geometric and energetic properties.

Contribution

It introduces a novel approach that generalizes previous work on logarithmic energy to Riesz energies and separation distances for zeros of elliptic polynomials.

Findings

Derived formulas for expected Riesz energies of zeros

Computed expected separation distances between zeros

Extended previous results to more general energy functions

Abstract

In 2011, Armentano, Beltr\'an and Shub obtained in \cite{ABS11} a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.