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arXiv · 2609.15984

Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots

Hongkai Zhu
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Abstract

Let $X_1,X_2,\ldots$ be i.i.d. complex-valued random variables with arbitrary Borel probability law $μ$, and set $P_n(z)=\prod_{j=1}^n(z-X_j)$. For every deterministic sequence $k_n=o(n)$, we prove that the empirical zero measure of the $k_n$-th derivative $P_n^{(k_n)}$ converges weakly to $μ$ almost surely.

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Hongkai Zhu. 2026-09-14. Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots. https://arxiv.org/abs/2609.15984

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