A Stability Version of the Gauss-Lucas Theorem and Applications

2018/05/26 by Stefan Steinerberger, Steinerberger, Stefan
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV

paper · pdf · doi:10.48550/arxiv.1805.10454

to appear in Journal of the Australian Mathematical Society

arxiv created 2018/12/14 · arxiv updated 2018/12/18

Abstract

Let p:ℂ → ℂ be a polynomial. The Gauss-Lucas theorem states that its critical points, p'(z) = 0, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose p has n+m roots where n are inside the unit disk, max_{1 ≤ i ≤ n}|a_i| ≤ 1, and m are outside min_{n+1 ≤ i ≤ n+m} |a_i| ≥ d > 1 + (2 m)/(n), then p' has n-1 roots inside the unit disk and m roots at distance at least (dn - m)/(n+m) > 1 from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for n sufficiently large each of the m roots has a critical point at distance ∼ n^-1.

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