Topological Triviality of $μ$-constant Deformations of Type f(x) + tg(x)

Abstract

We show that every $μ$-constant family of isolated hypersurface singularities of type f(x) + tg(x), where t is a parameter, is topologically trivial. In the proof we construct explicitely a vector field trivializing the family. The proof uses only the curve selection lemma and hence, for an appropriately translated statement, also works over the reals. Some applications to the study of singularities at infinity of complex polynomials are given.