The Le numbers of the square of a function and their applications

Lê numbers were introduced by Massey with the purposeof numerically controlling the topological properties of familiesof non-isolated hypersurface singularities and describing thetopology associated with a function with non-isolated singularities.They are a generalization of the Milnor number for isolatedhypersurface singularities. In this note the authors investigatethe composite of an arbitrary square-free f and z². They geta formula for the Lê numbers of the composite, and considertwo applications of these numbers. The first application isconcerned with the extent to which the Lê numbers areinvariant in a family of functions which satisfy some equisingularitycondition, the second is a quick proof of a new formula forthe Euler obstruction of a hypersurface singularity. Severalexamples are computed using this formula including any X definedby a function which only has transverse D(q, p) singularitiesoff the origin.