Multivariable CLT for critical points

We prove a multivariate central limit theorem for the numbers of critical points with all possible indexes which lie above a level of a non-necessarily isotropic Gaussian random field. We prove the non-degeneracy of the limit joint distribution in the isotropic case. We also consider the degenerate case when the value is not restricted to lie above a level. We extend, to the non-isotropic framework, known results by Estrade \& Le{\'o}n and Nicolaescu for the Euler characteristic of an excursion set and for the total number of critical points of Gaussian random fields. Though we use the classical tools of chaotic expansions and fourth moment theorem, our proof of the non-degeneracy of the limit distribution does not focus on the explicit description of the lowest order chaotic components but it transforms some convenient Hermite coefficients of arbitrary order into functions of the eigenvalues of a Gaussian Orthogonal Ensemble (GOE) random matrix and use the classical Laplace Method to conclude.