  The objective of this paper is to discuss invariants of singularities of
algebraic schemes over fields of positive characteristic, and to show how they
yield the simplification of singularities.
  We focus here on invariants which arise in an inductive manner, namely by
successive elimination of variables. When applied to hypersurface singularities
they lead us to a refinement of the notion of multiplicity.
  The main theorem proves that, under some numerical conditions expressed by
these invariants, singularities can be simplified by blowups at centers
prescribed by this refinement.
