The multiplicity of pairs of modules and hypersurface singularities
  This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a submodule of finite length of a free module F over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of M is not equal to F, is the multiplicity of M.   In this paper we apply this result to the study of stratification conditions on families of hypersurfaces, obtaining the first set of invariants giving necessary and sufficient conditions for the A_f condition for hypersurfaces with non-isolated singularities. 
Terence Gaffney