Leaky Roots and Stable Gauss-Lucas Theorems

Abstract

Let $p:\mathbb{C}\to \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. A recent quantitative version Totik shows that if almost all roots are contained in a bounded convex domain $K\subset \mathbb{C}$, then almost all roots of the derivative $p^{\prime }$ are in a $\epsilon -$neighborhood $K_{\epsilon }$ (in a precise sense). We prove another quantitative version: if a polynomial $p$ has $n$ roots in $K$ and $\lesssim c_{K,\epsilon }(n/\log n)$ roots outside of $K$, then $p^{\prime }$ has at least $n-1$ roots in $K_{\epsilon }$. This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture.