A well-known result, due to Ostrowski, states that if
${{\left\| P-Q \right\|}_{2}}\,<\,\varepsilon $, then the roots
$({{x}_{j}})$ of
$P$ and
$({{y}_{j}})$ of
$Q$ satisfy
$\left| {{x}_{j}}\,-\,{{y}_{j}} \right|\,\le \,Cn{{\varepsilon }^{1/n}}$, where
$n$ is the degree of
$P$ and
$Q$. Though there are cases where this estimate is sharp, it can still be made more precise in general, in two ways: first by using Bombieri’s norm instead of the classical
${{l}_{1}}$ or
${{l}_{2}}$ norms, and second by taking into account the multiplicity of each root. For instance, if
$x$ is a simple root of
$P$, we show that
$\left| x\,-\,y \right|\,<\,C\varepsilon $ instead of
${{\varepsilon }^{1/n}}$. The proof uses the properties of Bombieri’s scalar product andWalsh Contraction Principle.