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How the Roots of a Polynomial Vary with its Coefficients: A Local Quantitative Result
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 1999
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