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Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots
Abstract
Let pn be a random polynomial of degree n whose roots are independent and identically distributed according to a rotationally invariant probability measure μ0 on the complex plane with finite logarithmic moment. If kn/n→ t∈(0,1), we prove that, as n → ∞, the empirical zero measure of the kn-th derivative of pn 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 μ0. This proves a conjecture of Hoskins and Kabluchko [Exp. Math. 32 (2023), no. 4].
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BibTeX
@misc{orourke2026repeated,
title = {{Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots}},
author = {Sean O'Rourke},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23909},
eprint = {2609.23909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23909},
}
RIS
TY - UNPB
TI - Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots
AU - Sean O'Rourke
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.23909
UR - https://arxiv.org/abs/2609.23909
AB - 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→ t∈(0,1), we prove that, as n → ∞, 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].
ER -