A self contained proof of the Rice formula for random fields Jean-Marc Azais Summary: A self contained proof of the Rice formula for random fields Jean-Marc 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 so-called 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. Collections: Mathematics