Almost sure behavior of the zeros of iterated derivatives of random polynomials

2023/07/13 by Marcus Michelen, Michelen, Marcus, Xuan-Truong Vu +1 · 2 citations
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.2307.06788

openalex publication_date 2023/07/13 · openalex created_date 2023/07/15 · openalex updated_date 2026/07/28

Abstract

Let Z₁, Z₂,… be independent and identically distributed complex random variables with common distribution μ and set P_n(z) := (z - Z₁)⋯ (z - Z_n) . Recently, Angst, Malicet and Poly proved that the critical points of P_n converge in an almost-sure sense to the measure μ as n tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed k∈ ℕ, the empirical measure of zeros of the kth derivative of P_n converges to μ in the almost sure sense, as conjectured by Angst-Malicet-Poly.

Cited by

• Critical points of random polynomials and finite free cumulants
• Dynamics of rotationally invariant polynomial root sets under iterated differentiations

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