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

2022/12/22 by Marcus Michelen, Michelen, Marcus, Xuan-Truong Vu +1
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2212.11867

openalex publication_date 2022/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X₁,X₂,… be independent and identically distributed random variables in ℂ chosen from a probability measure μ and define the random polynomial P_n(z)=(z-X₁)…(z-X_n) . We show that for any sequence k = k(n) satisfying k ≤ log n / (5 loglog n), the zeros of the kth derivative of P_n are asymptotically distributed according to the same measure μ. This extends work of Kabluchko, which proved the k = 1 case, as well as Byun, Lee and Reddy who proved the fixed k case.

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