  We consider the problem of efficient integration of an n-variate polynomial
with respect to the Gaussian measure in R^n and related problems of complex
integration and optimization of a polynomial on the unit sphere. We identify a
class of n-variate polynomials f for which the integral of any positive integer
power f^p over the whole space is well-approximated by a properly scaled
integral over a random subspace of dimension O(log n). Consequently, the
maximum of f on the unit sphere is well-approximated by a properly scaled
maximum on the unit sphere in a random subspace of dimension O(log n). We
discuss connections with problems of combinatorial counting and applications to
efficient approximation of a hafnian of a positive matrix.
