Relative (?)-determinacy Of Relative Smooth Map-germs

The determinacy of smooth map-germ is an important subject of singularity. J.N.Mather worked initially on this topic. Then Radmila Bulajich brought forward the concept of relative map-germ, and solved the problem of relative versality for relative map-germs, and got the theorem of versal unfoldings of relative map-germs. Particularly, in this paper, we mainly consider this condition: S = RS x {0} , T = 0. We denote by G(n,s;m,0) the set of relative smooth map-germs from (Rn,Rs) to (Rm,0), then we can consider T AES,0(f) and TAs,0(f) , by which denote the tangent space at f and the orbit tangent space of ε(n,s;m,0) under the action of group As,0 respectively.With the aid of the relative Malgrange preparation theorem of reference [4] and the methods of reference [5], I try to discuss the relative finite A -determinacy of relative map-germs under the action of the group As,0 through threechapters.In the first chapter, 1 introduce briefly the former experts work of finite determinacy of map-germs and the background of this paper.In the second chapter, firstly, I present some marks and definitions, then discuss several le.mmas that closely relate to the primary theorem of this paper. Successively, I introuduce relative Malgrange preparation theorem, the approach theorem of relative map-germs, and the approach theorem of relative map-germs with single parameter family and so on.In the last section, the main purpose is to prove the primary theorem. We conclude a sufficient and necessary condition of the relative finite A-determinacy of relative smooth map-germs, i. e. for each ∈ε(n,s;m,0) , f is finite As,0 -determinant if and only if for some natural number k,or...