Relative Determinacy Of Smooth Function Germs

Finite determinacy of smooth map germs is related to the most important local property—stability, and is one of important research area in singularity theory. In this paper, we mainly studied relative finite determinacy of smooth function germs with repect to the subgroups (?)_s and (?)_r(S) of (?), and make some discussions on infinite determinacy of function germs with respect to the above subgroups.This paper will be divided into five chapters.In the first chapter, we briefly introduce the development course of singularity theory, mainly researches and study trends. Meanwhile, we introduce our researches and innovations of this paper. In the second chapter, we introduce basic concepts and ideas in singularity theory and algebraic geometry, including ring of function germs, group of local diffeomorphisms, finite codimension, finite determinacy, radical ideals, algebraic set etc.The main parts are in the third, fouth and fifth chapter.In chapter 3, finite determinacy of function germs relative the subgroup (?)_s of (?) is studied, where S is an algebraic set germs in (R~n,0). The main result generalized some results of Kushner(92) and other ones in some sence, and can be used to get more sharp estimation sometimes.In chapter 4, we make further discussion on finite determinacy of function germs relative the subgroup (?)_r (S) of (?)_s. Since S is an general algebraic set germs, the main result is more general than Tang Tieqiao's results(2003).Finally, we make some discussion on infinite determinacy of function germs relative to the subgroups mentioned above, and the result generalized Wilson's theorem(81).