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

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