Leaky Roots and Stable Gauss-Lucas Theorems

TL;DR

This paper presents a new quantitative version of the Gauss-Lucas theorem, showing that if most roots of a polynomial are in a convex domain and few are outside, then most critical points are close to that domain.

Contribution

It proves a novel quantitative bound relating the number of roots outside a convex domain to the location of critical points, advancing understanding of polynomial root distributions.

Findings

Most critical points lie near the convex hull of roots under certain conditions

Established a bound involving the number of roots outside the domain

Connected results to a conjecture of the first author

Abstract

Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the convex hull of its roots. 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 $p'$ are in a $\varepsilon-$neighborhood $K_{\varepsilon}$ (in a precise sense). We prove another quantitative version: if a polynomial $p$ has $n$ roots in $K$ and $\lesssim c_{K, \varepsilon} (n/\log{n})$ roots outside of $K$, then $p'$ has at least $n-1$ roots in $K_{\varepsilon}$. 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.