  To prove that a polynomial is nonnegative on R^n one can try to show that it
is a sum of squares of polynomials (SOS). The latter problem is now known to be
reducible to a semidefinite programming (SDP) computation much faster than
classical algebraic methods, thus enabling new speed-ups in algebraic
optimization. However, exactly how often nonnegative polynomials are in fact
sums of squares of polynomials remains an open problem. Blekherman was recently
able to show that for degree k polynomials in n variables -- with k>=4 fixed --
those that are SOS occupy a vanishingly small fraction of those that are
nonnegative on R^n, as n tends to infinity.
  With an eye toward the case of small n, we refine Blekherman's bounds by
incorporating the underlying Newton polytope, simultaneously sharpening some of
his older bounds along the way. Our refined asymptotics show that certain
Newton polytopes may lead to families of polynomials where efficient SDP can
still be used for most inputs.
