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# How the Roots of a Polynomial Vary with Its Coefficients: A Local Quantitative Result

A well-known result, due to Ostrowski, states that if $\Vert P-Q \Vert_2< \varepsilon$, then the roots $(x_j)$ of $P$ and $(y_j)$ of $Q$ satisfy $|x_j -y_j|\le C n \varepsilon^{1/n}$, 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 $|x-y|< C \varepsilon$ instead of $\varepsilon^{1/n}$. The proof uses the properties of Bombieri's scalar product and Walsh Contraction Principle.