Schur studied limits of the arithmetic means 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 . We show that , 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 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 . Nous prouvons que . Nous donnons une estimation du taux de convergence, grâce à une généralisation d'un théorème de Erdős–Turán sur la distribution des zéros.
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Igor E. Pritsker 1
@article{CRMATH_2009__347_3-4_119_0, author = {Igor E. Pritsker}, title = {Means of algebraic numbers in the unit disk}, journal = {Comptes Rendus. Math\'ematique}, pages = {119--122}, publisher = {Elsevier}, volume = {347}, number = {3-4}, year = {2009}, doi = {10.1016/j.crma.2009.01.002}, language = {en}, }
Igor E. Pritsker. 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://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.002/
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