A Stability Version of the Gauss-Lucas Theorem and Applications

TL;DR

This paper presents a stability version of the Gauss-Lucas theorem, showing how the roots and critical points of a polynomial relate when roots are inside or outside the unit disk, with sharp constants and applications.

Contribution

It introduces a stability version of the Gauss-Lucas theorem with explicit bounds and sharp constants, extending classical results to polynomials with roots outside the unit disk.

Findings

Critical points remain close to roots outside the disk under certain conditions.

The number of roots inside the disk decreases by one for the derivative.

Roots outside the disk are at least a certain distance from the origin.

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. We prove a stability version whose simplest form is as follows: suppose $p$ has $n+m$ roots where $n$ are inside the unit disk, $$ \max_{1 \leq i \leq n}{|a_i|} \leq 1, \quad \mbox{and $m$ are outside} \quad \min_{n+1 \leq i \leq n+m}{ |a_i|} \geq d > 1 + \frac{2 m}{n},$$ 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 $\sim n^{-1}$.