  We prove that zeros and critical points of a random polynomial $p_N$ of
degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is
conditioned to have $p_N(\xi)=0$ for a fixed $\xi \in \C\backslash\set{0},$ we
prove that there is a unique critical point z in the annulus
$N^{-1-\ep}<\abs{z-\xi}< N^{-1+\ep}}$ and no critical points closer to $\xi$
with probability at least $1-O(N^{-3/2+3\ep}).$ We also prove an analogous
statement in the more general setting of random meromorphic functions on a
closed Riemann surface.
