A quantitative Gauss–Lucas theorem

Abstract

. 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 ε> 0 there is an α ε < 1 such that if a polynomial P n of degree at most n has k ≥ α ε n zeros in K , then P (cid:2) n has at least k − 1 zeros in the ε -neighborhood of K . Estimates are given for the dependence of α ε on ε .