Addendum to "Energies of zeros of random sections on Riemann surfaces" [arXiv:0705.2000]. Indiana Univ. Math. J. 57 (2008), no. 4, 1753-1780

TL;DR

This addendum clarifies how to compute the asymptotics of the expected energy of zeros of random polynomials on the Riemann sphere using the main results from a prior paper, correcting previous errors and aligning with other known results.

Contribution

It provides a rigorous method to calculate the asymptotics of expected zero energy on the Riemann sphere, correcting earlier ad-hoc approaches.

Findings

Corrected the asymptotic energy calculation for zeros of random polynomials

Aligned the results with those of Armentano-Beltran-Shub

Resolved previous calculation errors

Abstract

This is an addendum to the article of Qi Zhong cited above [arXiv:0705.2000]. It outlines how to apply the main result of that article to calculate the asymptotics of the expected energy of zeros of random polynomials on the Riemann sphere $S^2$ with respect to the log chordal distance $\log [z, w]$. The cited article did not calculate the asymptotic energy this way, but by an ad-hoc method, and the calculation contained some errors. The correct calculation here agrees (up to the stipulted remainder) with that of Armentano- Beltran-Shub.