Multijet bundles and application to the finiteness of moments for zeros of Gaussian fields

We define a notion of multijet for functions on ${\mathbb{R}}^n$, which extends the classical notion of jets in the sense that the multijet of a function is defined by contact conditions at several points. For all $p\ge 1$ we build a vector bundle of $p$-multijets, defined over a well-chosen compactification of the configuration space of $p$ distinct points in ${\mathbb{R}}^n$. As an application, we prove that the linear statistics associated with the zero set of a centered Gaussian field on a Riemannian manifold have a finite $p$-th moment as soon as the field is of class~${\mathcal{C}}^p$ and its $(p-1)$-jet is nowhere degenerate. We prove a similar result for the linear statistics associated with the critical points of a Gaussian field and those associated with the vanishing locus of a holomorphic Gaussian field.