  We study the minimization of fixed-degree polynomials over the simplex. This
problem is well-known to be NP-hard, as it contains the maximum stable set
problem in graph theory as a special case. In this paper, we consider a
rational approximation by taking the minimum over the regular grid, which
consists of rational points with denominator $r$ (for given $r$). We show that
the associated convergence rate is $O(1/r^2)$ for quadratic polynomials. For
general polynomials, if there exists a rational global minimizer over the
simplex, we show that the convergence rate is also of the order $O(1/r^2)$. Our
results answer a question posed by De Klerk et al. (2013) and improves on
previously known $O(1/r)$ bounds in the quadratic case.
