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The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-normof the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is boundedby $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot\|_\infty$ denotes the supremum norm on the polydisc $\D^n$. The main result ofthis paper is that this inequality is hypercontractive, i.e., the constant canbe taken to be $C^m$ for some $C>1$. Combining this improved version of theBohnenblust--Hille inequality with other results, we obtain the following: TheBohr radius for the polydisc $\D^n$ behaves asymptotically as $\sqrt{(\logn)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constantfor the set of frequencies $\bigl\{\log n: n \text{a positive integer} \leN\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as$N\to \infty$.

The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive