  2-point topological charge correlation functions of several types of
geometric singularity in gaussian random fields are calculated explicitly,
using a general scheme: zeros of $n$-dimensional random vectors, signed by the
sign of their jacobian determinant; critical points (gradient zeros) of real
scalars in two dimensions signed by the hessian; and umbilic points of real
scalars in two dimensions, signed by their index. The functions in each case
depend on the underlying spatial correlation function of the field. These
topological charge correlation functions are found to obey the first
Stillinger-Lovett sum rule for ionic fluids.
