  Let $\cP_n$ be the complex vector space of all polynomials of degree at most
$n$. We give several characterizations of the linear operators $T\in\cL(\cP_n)$
for which there exists a constant $C > 0$ such that for all nonconstant
$p\in\cP_n$ there exist a root $u$ of $p$ and a root $v$ of $Tp$ with
$|u-v|\leq C$. We prove that such perturbations leave the degree unchanged and,
for a suitable pairing of the roots of $p$ and $Tp$, the roots are never
displaced by more than a uniform constant independent on $p$. We show that such
``good'' operators $T$ are exactly the invertible elements of the commutative
algebra generated by the differentiation operator. We provide upper bounds in
terms of $T$ for the relevant constants.
