Finite R_r(S;n)-Determinacy Of Smooth Function Germs

Finite deterrainacy of smooth map germs is one of im-portant special topics in the theory of singularity. As for finite determinacy of smooth function germs, the essential form is finite determinacy; later, finite (S;n)-deter-minacy and finite 5Rr-determinacy were generalized from finite -determinacy. Professor Li had discussed finite r-determinacy of smooth map germs in [2], and studied finite -determinacy of smooth function germs as a special topic in [3] ; Porto and Loibel had discussed finite (s;n)-determinacy and finite (s;n)-determinacy of smooth funct-ion germs in [1]. In the paper, the author has discussed finite (S;n)-Aeterminacy of smooth function germs on the basis of the former method, and he has obtained some results as follows:( i ) The sufficient and necessary condition forrecognition of that a function germ is (S;n)-trivial;(ii) The sufficient and necessary condition for recognition of that a function germ is finitely (S;n)-determined;(iii) The relationship between the finite determi-inacy and the finite (s;n)-determinacy of smooth function germs.