高阶Morse芽的存在性

EXISTENCE OF HIGHER ORDER MORSE GERMS

本文研究了多元C∞函数芽环中高阶Morse芽的存在性问题.利用由函数芽的偏导数生成的理想和C∞函数芽上的右等价关系,获得了在C∞函数芽环中,除了二元C∞函数芽环中有三阶和四阶的Morse芽以后,不再存在其它的Morse芽.以致在三元以上的C∞函数芽环中Morse引理不能推广到较高阶的情形.

In this paper,it is discussed whether there exist higher order Morse germs in the ring of C^∞ functions germs of several variables. Using the idea generated by partial derivatives of function and right equivalence of functions, it is obtained that there are no longer higher order Morse germs in the ring of C^∞ functions germs of several variables except there are three order and four order Morse germs in the ring of C^∞ functions germs of two variables. So Morse Lemma can＇t be generalized to higher order in the ring of C^∞ functions germs of several variables.