A quantitative Gauss–Lucas theorem

A conjecture of T. Richards is proven which yields a quantitative version of the classical Gauss–Lucas theorem: if $K$ is a convex set, then for every $\varepsilon \gt 0$ there is an $\alpha_{\varepsilon} \lt 1$ such that if a polynomial $P_n$ of degree at most $n$ has $k \geq \alpha_{\varepsilon} n$ zeros in $K$, then $P^{\prime}_n$ has at least $k-1$ zeros in the $\varepsilon$-neighborhood of $K$. Estimates are given for the dependence of $\alpha_{\varepsilon}$ on $\varepsilon$.

Keywords

zeros, critical points, Gauss–Lucas theorem, Cauchy transform, potential theory

2010 Mathematics Subject Classification

26C10, 31A15

Full Text (PDF format)

Received 6 February 2021

Accepted 9 August 2021

Published 16 May 2022