  The asymptotic Samuel function generalizes to arbitrary rings the usual order
function of a regular local ring. In this paper, we use this function to
introduce the notion of the Samuel slope of a Noetherian local ring, and we
study some of its properties. In particular, we focus on the case of a local
ring at singular point of a variety, and, among other results, we prove that
the Samuel slope of these rings is related to some invariants used in
algorithmic resolution of singularities.
