[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.
Accepté le :
Publié le :
@article{CRMATH_2008__346_5-6_267_0,
author = {Erd\'elyi, Tam\'as},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.01.020/}
}
TY - JOUR AU - Erdélyi, Tamás TI - An improvement of the Erdős–Turán theorem on the distribution of zeros of polynomials JO - Comptes Rendus. Mathématique PY - 2008 SP - 267 EP - 270 VL - 346 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.01.020/ DO - 10.1016/j.crma.2008.01.020 LA - en ID - CRMATH_2008__346_5-6_267_0 ER -
%0 Journal Article %A Erdélyi, Tamás %T An improvement of the Erdős–Turán theorem on the distribution of zeros of polynomials %J Comptes Rendus. Mathématique %D 2008 %P 267-270 %V 346 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.01.020/ %R 10.1016/j.crma.2008.01.020 %G en %F CRMATH_2008__346_5-6_267_0
Erdélyi, Tamás. An improvement of the Erdős–Turán theorem on the distribution of zeros of polynomials. Comptes Rendus. Mathématique, Tome 346 (2008) no. 5-6, pp. 267-270. doi : 10.1016/j.crma.2008.01.020. http://www.numdam.org/articles/10.1016/j.crma.2008.01.020/
[1] Discrepancy of Signed Measures and Polynomial Approximation, Springer, New York, 2002
[2] Littlewood-type problems on , Proc. London Math. Soc. (3), Volume 79 (1999), pp. 22-46
[3] On the distribution of roots of polynomials, Ann. Math. (1950), pp. 105-119
[4] On the first and second main theorem in Turán's theory of power sums (Erdős, P., ed.), Studies in Pure Mathematics, Birkhäuser Verlag, Basel, 1983, pp. 259-269 (To the Memory of Paul Turán)
[5] Inequalities for polynomials with a prescribed zero, Math. Z., Volume 168 (1979), pp. 105-116
[6] Review on the book “Discrepancy of Signed Measures and Polynomial Approximation”, Bull. London Math. Soc., Volume 35 (2003), pp. 284-287
[7] V. Totik, P. Varjú, Polynomials with prescribed zeros and small norm, Acta Sci. Math. (Szeged), in press
Cité par Sources :
