  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 $\sim n^{-1}$.
