摘要
设Ｅｎ是在０∈Ｒ~ｎ的Ｃ函数芽环，Ｍ是Ｅ_ｎ中唯一的极大理想．如果ｆ∈Ｍ~２且其二阶Ｈｅｓｓａｉｎ是非退化的，则ｆ同构于它的二阶Ｈｅｓｓａｉｎ，这就是著名的Ｍｏｒｓｅ引理，本文将讨论两个变元的Ｃ~∞函数芽，得到：（１）若ｆ∈Ｍ~３∈Ｅ_(ｘｙ)，且其三阶Ｈｅｓｓａｉｎ是非退化的，则ｆ同构于它的三阶Ｈｅｓｓａｉｎ。 （２）若ｆ∈Ｍ_４∈Ｅ_(ｘｙ)，其四阶Ｈｅｓｓａｉｎ是非退化的，则ｆ同构于它的四阶Ｈｅｓｓａｉｎ．显然，这是Ｍｏｒｓｅ引理的一种推广．
Assume that E_n is the ring of the C~∞ function germs at 0∈ R^n,M is the unique maximal ideal in E_n. If f ∈M^2 and its quadratic Hessain is non-degenerate, then f is isomorphic to its quadratic Hessain. This is famous Morse lemma. In this paper, We will discuss C funtion germs in two variables. The results show that (1) If f∈M^3(E_(xy) and its cubic Hessain is non-degenerate, then f is isomorphic to its cubic Hessain. (2) If f∈M^4( M E_(xy) and its Hessain of degree 4 is non-degenerate,then f is isomorphic to its Hessain of degree 4. Obviously, this is a generalization of Morse lemma.
