On the Distribution of Critical Points of a Polynomial

TL;DR

This paper demonstrates that the empirical distribution of critical points of polynomials with roots chosen i.i.d. from a measure on the unit circle converges weakly to that measure.

Contribution

It proves the weak convergence of the critical points' empirical distribution to the original measure for polynomials with roots on the unit circle.

Findings

Critical points' distribution converges to measure μ

Weak convergence holds for i.i.d. roots on the circle

Results extend understanding of polynomial critical points

Abstract

This paper proves that if points $Z_1,Z_2,...$ are chosen independently and identically using some measure $\mu$ from the unit circle in the complex plane, with $p_n(z) = (z-Z_1)(z-Z_2)...(z-Z_n)$, then the empirical distribution of the critical points of $p_n$ converges weakly to $\mu$.