Year: 2026
Type: preprint
Abstract: Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure $μ_0$ on the complex plane with finite logarithmic moment. If $k_n/n\to t\in(0,1)$, we prove that, as $n \to \infty$, the empirical zero measure of the $k_n$-th derivative of $p_n$ converges weakly in probability to a deterministic rotationally invariant probability measure. We describe the limiting measure explicitly in terms of the radial quantile function of $μ_0$. This proves a conjecture of Hoskins and Kabluchko [Exp. Math. 32 (2023), no. 4].
Source: arXiv (Cornell University)
Author: O'Rourke, Sean
Institutions: University of Colorado Boulder, University of Colorado System
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7214042735