Abstract:If $p:\mathbb{C} \to \mathbb{C}$ 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 $\mu$ 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 $\mu$ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.
From: Sean O'Rourke [view email]
[v1]
Sat, 14 Sep 2024 21:39:50 UTC (78 KB)
[v2]
Mon, 29 Jun 2026 20:01:12 UTC (82 KB)
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