Zeros of a growing number of derivatives of random polynomials with independent roots

TL;DR

This paper proves that for a broad class of random polynomials with i.i.d. roots, the zeros of their derivatives up to a certain growing order are asymptotically distributed according to the same measure as the roots, extending previous fixed-order results.

Contribution

It extends prior work by showing that the zeros of the $k$th derivative of random polynomials with i.i.d. roots follow the original root distribution for growing $k$ up to a logarithmic scale.

Findings

Zeros of derivatives are asymptotically distributed as the original roots.

The result holds for derivatives of order up to roughly $rac{ ext{log } n}{ ext{log log } n}$.

Generalizes previous fixed-order derivative zero distribution results.

Abstract

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $\mu$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $\mu$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.