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Let $f$ be a Gaussian random field on $\mathbb{R}^d$ and let $X$ be thenumber of critical points of $f$ contained in a compact subset. A long-standingconjecture is that, under mild regularity and non-degeneracy conditions on $f$,the random variable $X$ has finite moments. So far, this has been establishedonly for moments of order lower than three. In this paper, we prove theconjecture. Precisely, we show that $X$ has finite moment of order $p$, as soonas, at any given point, the Taylor polynomial of order $p$ of $f$ isnon-degenerate. We present a simple and general approach that is not specificto critical points and we provide various applications. In particular, we showthe finiteness of moments of the nodal volumes and the number of criticalpoints of a large class of smooth, or holomorphic, Gaussian fields, includingthe Bargmann-Fock ensemble.

The number of critical points of a Gaussian field: finiteness of moments