  The asymptotic Samuel function generalizes to arbitrary rings the usual order
function of a regular local ring. Here we explore some natural properties in
the context of excellent, equidimensional rings containing a field. In
addition, we establish some results regarding the Samuel Slope of a local ring.
This is an invariant related with algorithmic resolution of singularities of
algebraic varieties. Among other results, we study its behavior after certain
faithfully flat extensions.
