[Un raffinement du théorème de Erdős–Turán sur la distribution des zéros de polynômes]
Nous prouvons un raffinement délicat d'un résultat classique de P. Erdős et P. Turán sur la distribution des zéros de polynômes. Bien que notre preuve soit brève et plutôt élémentaire, elle nous permet d'obtenir comme corollaire et sans recourir à la théorie du potentiel, une amélioration d'un résultat récent et élégant de V. Totik et P. Varjú : si le module sur le cercle unité du plan complexe d'un polynôme P monique, de degré n et à coefficients complexes est uniformément au plus , alors la multiplicité de chaque zéro de P sur le cercle unité est . Notre approche repose sur l'observation, à notre avis intéressante, que le théorème de Erdős–Turán peut en quelque sorte s'auto-raffiner.
We prove a subtle ‘one-sided’ improvement of a classical result of P. Erdős and P. Turán on the distribution of zeros of polynomials. The proof of this improvement is quite short and rather elementary. Nevertheless it allows us to obtain a beautiful recent result of V. Totik and P. Varjú as a simple corollary, and in a somewhat stronger form, without any use of a potential theoretic machinery. Namely, if the modulus of a monic polynomial P of degree n (with complex coefficients) on the unit circle of the complex plane is at most uniformly, then the multiplicity of each zero of P on the unit circle is . Our approach is based on the interesting observation that the Erdős–Turán Theorem improves itself.
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@article{CRMATH_2008__346_5-6_267_0,
author = {Tam\'as Erd\'elyi},
title = {An improvement of the {Erd\H{o}s{\textendash}Tur\'an} theorem on the distribution of zeros of polynomials},
journal = {Comptes Rendus. Math\'ematique},
pages = {267--270},
publisher = {Elsevier},
volume = {346},
number = {5-6},
year = {2008},
doi = {10.1016/j.crma.2008.01.020},
language = {en},
}
Tamás Erdélyi. An improvement of the Erdős–Turán theorem on the distribution of zeros of polynomials. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 267-270. doi : 10.1016/j.crma.2008.01.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.020/
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