Means of algebraic numbers in the unit disk

Pritsker, Igor E.

doi:10.1016/j.crma.2009.01.002

Number Theory

Means of algebraic numbers in the unit disk

Presented by: Jean-Pierre Serre

Pritsker, Igor E.¹

¹ Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 119-122.

Abstract
Résumé

Schur studied limits of the arithmetic means $s_{n}$ of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that ${lim sup}_{n \to \infty} | s_{n} | ⩽ 1 - \sqrt{e} / 2$ . We show that $s_{n} \to 0$ , and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros.

Schur a étudié les limites des moyennes arithmétiques $s_{n}$ des zéros pour les polynômes à coefficients entiers de degré n ayant des zéros simples dans le disque unité fermé. Lorsque les coefficients dominants restent bornés, Schur a démontré que ${lim sup}_{n \to \infty} | s_{n} | ⩽ 1 - \sqrt{e} / 2$ . Nous prouvons que $s_{n} \to 0$ . Nous donnons une estimation du taux de convergence, grâce à une généralisation d'un théorème de Erdős–Turán sur la distribution des zéros.

Received: 2008-11-19
Accepted: 2009-01-09
Published online: 2009-02-01

DOI: 10.1016/j.crma.2009.01.002

Author's affiliations:

Pritsker, Igor E. ¹

¹ Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

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Pritsker, Igor E. Means of algebraic numbers in the unit disk. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 119-122. doi : 10.1016/j.crma.2009.01.002. https://www.numdam.org/articles/10.1016/j.crma.2009.01.002/

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