2024/09/14 by Sean O’Rourke, O'Rourke, Sean, Noah Williams +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #Probability (math.PR) #Probability and Statistical Research
paper · pdf · doi:10.48550/arxiv.2409.09538
openalex publication_date 2024/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If p:ℂ → ℂ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when p is a random polynomial of degree n with roots chosen independently from a radially symmetric, compactly supported probability measure μ in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of p. This allows us to compute the asymptotic fluctuations of the largest critical points as the degree n tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of μ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.