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

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 \begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*} 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 [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed $k$ case.