Almost sure behavior of the zeros of iterated derivatives of random polynomials

TL;DR

This paper proves that for random polynomials with i.i.d. roots, the zeros of their derivatives converge almost surely to the same distribution as the roots themselves, extending previous results to higher derivatives.

Contribution

It establishes the almost sure convergence of zeros of the $k$th derivatives of random polynomials to the root distribution, confirming a conjecture for all fixed derivatives.

Findings

Zeros of derivatives converge to the root distribution almost surely

Extension of previous convergence results to higher derivatives

Confirms conjecture by Angst-Malicet-Poly

Abstract

Let $Z_1,\, Z_2,\dots$ be independent and identically distributed complex random variables with common distribution $\mu$ and set $$ P_n(z) := (z - Z_1)\cdots (z - Z_n)\,. $$ Recently, Angst, Malicet and Poly proved that the critical points of $P_n$ converge in an almost-sure sense to the measure $\mu$ as $n$ tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed $k\in \mathbb{N}$, the empirical measure of zeros of the $k$th derivative of $P_n$ converges to $\mu$ in the almost sure sense, as conjectured by Angst-Malicet-Poly.