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The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive Defant, Andreas Frerick, Leonhard Ortega Cerdà, Joaquim Ounaïes, Myriam Seip, Kristian Funcions de diverses variables complexes Funcions holomorfes Funcions de variables complexes Functions of several complex variables Holomorphic functions Functions of complex variables The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$ -norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$. 2013-03-22T12:26:18Z 2013-03-22T12:26:18Z 2011 2013-03-22T12:26:18Z info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Reproducció del document publicat a: http://dx.doi.org/10.4007/annals.2011.174.1.13 Annals of Mathematics, 2011, vol. 174, num. 1, p. 485-497 http://dx.doi.org/10.4007/annals.2011.174.1.13 (c) Annals of Mathematics, 2011 info:eu-repo/semantics/openAccess Princeton University Press Articles publicats en revistes (Matemàtiques i Informàtica)