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$L^p$-norms and Mahler's measure of polynomials on the $n$-dimensional torus

We prove Nikol'skii type inequalities which for polynomials on the $n$-dimensional torus $\mathbb{T}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0 <p <q < \infty$). Among other things we show that $C=\sqrt{q/p}$ is the best constant such that $\|P\|_{L^q}\leq C^{\text{deg}(P)} \|P\|_{L^p}$ for all homogeneous polynomials $P$ on $\mathbb{T}^n$. We also prove an exact inequality between the $L^p$-norm of a polynomial $P$ on $\mathbb{T}^n$ and its Mahler measure $M(P)$, which is the geometric mean of $|P|$ with respect to the normalized Lebesgue measure on $\mathbb{T}^n$. Using extrapolation we transfer this estimate into a Khintchine-Kahane type inequality, which, for polynomials on $\mathbb{T}^n$, relates a certain exponential Orlicz norm and Mahler's measure. Applications are given, including some interpolation estimates.