On Zeroes of Random Polynomials and Applications to Unwinding

2018/07/15 by Stefan Steinerberger, Hau‐Tieng Wu, Steinerberger, Stefan +1 · 2 citations
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1807.05587

openalex publication_date 2018/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let μ be a probability measure in ℂ with a continuous and compactly supported density function, let z₁, …, z_n be independent random variables, z_i ∼ μ, and consider the random polynomial p_n(z) = ∏_k=1ⁿ(z - z_k). We determine the asymptotic distribution of \z ∈ ℂ: p_n(z) = p_n(0)\. In particular, if μ is radial around the origin, then those solutions are also distributed according to μ as n → ∞. Generally, the distribution of the solutions will reproduce parts of μ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

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