  We develop techniques for computing zeta functions associated with nilpotent
groups, not necessarily associative algebras, and modules, as well as
Igusa-type zeta functions. At the heart of our method lies an explicit
convex-geometric formula for a class of $p$-adic integrals under non-degeneracy
conditions with respect to associated Newton polytopes. Our techniques prove to
be especially useful for the computation of topological zeta functions
associated with algebras, resulting in the first systematic investigation of
their properties.
