Contact Finite Determinacy of Relative Map Germs

Abstract

The strong contact finite determinacy of relative map germs is studied by means of classical singularity theory. We first give the definition of a strong relative contact equivalence (or $\mathcal{K}_{S,T}$ equivalence) and then prove two theorems which can be used to distinguish the contact finite determinacy of relative map germs, that is, $f$ is finite determined relative to $\mathcal{K}_{S,T}$ if and only if there exists a positive integer $k$, such that $\mathcal{M}^k (n)Ԑ(S; n)^p ⊂ T\mathcal{K}_{S,T}(f)$.