 
Summary: A self contained proof of the Rice formula for random fields
JeanMarc Aza¨is
, azais@cict.fr Mario Wschebor
, wschebor@cmat.edu.uy
December 22, 2006
AMS subject classification: Primary 60G70 Secondary 60G15
Key words and phrases: Gaussian fields, Rice Formula,
After an elementary proof of the Area Formula, we give a proof of Rice Formula for the
expectation of the number of roots of a random system of equations. We provide a complete
proof which is new and quite elementary, and in any case shorter than previous ones (see for
example [1]).
Similar formulae hold true for higher order factorial moments of the number of roots (The
orem 2). Theorem 3 provides a formula for the expectation of the total weight, when random
weights are put in each root.
1 The Area formula
We begin with a proof of the socalled Area formula, under conditions that will be sufficient for
our main purpose. One can find this formula in its full generality in Federer [3] Th 3.2.5
For any function f, we denote Nf
u (T) the number of roots of the equation f(t) = u that
belong to the subset T of the domain of f.
