  We study dimensions of the faces of the cone of nonnegative polynomials and
the cone of sums of squares; we show that there are dimensional differences
between corresponding faces of these cones. These dimensional gaps occur in all
cases where there exist nonnegative polynomials that are not sums of squares.
As either the degree or the number of variables grows the gaps become very
large, asymptotically the gaps approach the full dimension of the vector space
of polynomials in $n$ variables of degree 2d. The gaps occur generically, they
are not a product of selecting special faces of the cones. Using these
dimensional differences we show how to derive inequalities that separate
nonnegative polynomials from sums of squares; the inequalities will hold for
all sums of squares, but will fail for some nonnegative polynomials.
