Function Bud Orbital Space C - Equivalent Properties And Applications

In the singularity theory,J.N.Mather proved the following famous theorem:Let f∈En.If there exists positive integer k such that ,then the germ f is determined.i.e. f∈En and ,then ,if , g is right equivalent to f .However,for the family of function germs , Nt xy can not be determined by using J.N.Mather theorem.But (,)1Nt xy is really right equivalent toIn this paper,starting from the concept of C ?equivalence of finite generated ideals in E n,we prove the following theorems:Theorem 1. and small enough then ,if and only ifBy applying theorem 1,we further prove theorem 2.If n and satisfies then g is right equivalent to f ,provided that g ? f is in M k and that Pnk is small enough.The above two theorems expose the relationship between C ?equivalence of orbit tangent spaces and R equivalence of germs while theorem 2 is then a generalization of J.N.Mather theorem.Finally,by practical computation we show that theorem 2 is efficient for determining of R ?equivalence of (,) Moreover,we also apply similar methods to discuss the proof of Morse lemma.