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

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 kth 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. [ABSTRACT FROM AUTHOR]