Smooth Function Germs Limited Decisive

The finite determinacy of smooth mapping germs is an important topic in singularity theory. In this paper, we carry out a research into the special case, that is the finite determinacy of smooth function germs under various subgroups of right equivalent group. We also discuss the finite determinacy relative to right equivalent group of smooth function germs under the operation of function germs. Otherwise, we generalize the Malgrange preparation theorem of which is very important to the local analysis into a certain form.The elementary thought of this paper is use the view of algebra to tackle the problems of analysis. This paper has six chapters. In the first chapter, we recall the historical background and recent development of singularity theory. In chapter two, we present the fundamental knowledge we need in our article. Chapter three introduce the recent results of finite determinacy of smooth function germs, contains R_r-determinacy, R(S;n)-determinacy, R,(S;n)-determinacy, R_s~*(n)- determinacy and the determinacy on algebraic sets. In chapter four, we generalize part results of chapter three and get some results more generally. In chapter five, we give some results for finite determinacy relative to right equivalent group of smooth function germs under the operation of function germs. Chapter five give a generalization of Malgrange preparation theorem.