  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 $p \geq 1$ 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.
