  We calculate bounds for orthant probabilities for the equicorrelated
multivariate normal distribution and use these bounds to show the following:
for degree $k>4$, the probability that a $k$-homogeneous polynomial in $n$
variables attains a relative maximum on a vertex of the $n$-dimensional simplex
tends to one as the dimension $n$ grows. The bounds we obtain for the orthant
probabilities are tight up to $\log(n)$ factors.
