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We prove Nikol’skii type inequalities that, for polynomials on the -dimensional torus , relate the -norm with the -norm (with respect to the normalized Lebesgue measure and ). Among other things, we show that  is the best constant such that  for all homogeneous polynomials  on . We also prove an exact inequality between the -norm of a polynomial  on  and its Mahler measure (), which is the geometric mean of || with respect to the normalized Lebesgue measure on . Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on , relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.

$$L^{p}$$ L p -Norms and Mahler’s Measure of Polynomials on the n-Dimensional Torus