  Critical points of a scalar quantitiy are either extremal points or saddle
points. The character of the critical points is determined by the sign
distribution of the eigenvalues of the Hessian matrix. For a two-dimensional
homogeneous and isotropic random function topological arguments are sufficient
to show that all possible sign combinations are equidistributed or with other
words, the density of the saddle points and extrema agree. This argument breaks
down in three dimensions. All ratios of the densities of saddle points and
extrema larger than one are possible. For a homogeneous Gaussian random field
one finds no longer an equidistribution of signs, saddle points are slightly
more frequent.
