  We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge
1$) converging to the global minimum of a polynomial $f$ on simple sets like
the unit hypercube. The novelty with respect to the converging sequence of
upper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets and
polynomial optimization, SIAM J. Optim. 21, pp. 864--885, 2010] is that only
elementary computations are required. For optimization over the hypercube, we
show that the new bounds $f^H_k$ have a rate of convergence in $O(1/\sqrt
{k})$. Moreover we show a stronger convergence rate in $O(1/k)$ for quadratic
polynomials and more generally for polynomials having a rational minimizer in
the hypercube. In comparison, evaluation of all rational grid points with
denominator $k$ produces bounds with a rate of convergence in $O(1/k^2)$, but
at the cost of $O(k^n)$ function evaluations, while the new bound $f^H_k$ needs
only $O(n^k)$ elementary calculations.
