  We consider the unconstrained optimization of multivariate trigonometric
polynomials by the sum-of-squares hierarchy of lower bounds. We first show a
convergence rate of $O(1/s^2)$ for the relaxation with degree $s$ without any
assumption on the trigonometric polynomial to minimize. Second, when the
polynomial has a finite number of global minimizers with invertible Hessians at
these minimizers, we show an exponential convergence rate with explicit
constants. Our results also apply to minimizing regular multivariate
polynomials on the hypercube.
