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

Beauzamy, Bernard

Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 3-12

Résumé

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.

Détail

DOI : 10.4153/CMB-1999-001-6

Mots-clés : 30C10