Computation Of Finite Codimensional Ideals For The Ring Of Gearms Of C~∞ Functions

In the singularity theory , the algebraic conditions of finite k—determined and universal deformation have been given by J.N.Mather. These conditions deal with an important issue : the computation of codimension of finite codimensional ideals in E_n. However , A key point of applied these theories is that how to convert these abstract algebraic conditions into a concrete algorithm. Which is often a difficulty in the application of these theories.In this thesis ,by means of some algebraic knowledges and tricks of Nakayama Lemma , we will prove : (1) The generators of finite codimensional ideals in E_n can be simplified to polynomials or monomials about the relational theorems and methods. (2) Based on simplify of the generators and by means of a direct sum decomposition of E_n : E_n = P_n⁰ +P_n¹+…+P_n^k+?M(n)^k+1. We will propose some methods and principles about computation of codimension of finite codimensional ideals in E_n, and apply the relational results into the computation of finite k—determined and universal deformation of germs of finite codimension and Malgrange preparation theorem. The pratical examples demonstrate that our methods are efficient and have wider adaptability.