A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

We study the probability distribution F(u) of the maximum of smooth Gaussian fields defined on compact subsets of having some geometric regularity. Our main result is a general expression for the density of F. Even though this is an implicit formula, one can deduce from it explicit bounds for the density, and hence for the distribution, as well as improved expansions for 1-F(u) for large values of u. The main tool is the Rice formula for the moments of the number of roots of a random system of equations over the reals. This method enables also to study second-order properties of the expected Euler characteristic approximation using only elementary arguments and to extend these kinds of results to some interesting classes of Gaussian fields. We obtain more precise results for the "direct method" to compute the distribution of the maximum, using the spectral theory of GOE random matrices.