A Stability Version of the Gauss-Lucas Theorem and Applications

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 $\sim n^{-1}$.