两类二元函数芽的一个共同性质及其应用
A COMMON PROPERTY OF TWO TYPES OF FUNCTION GERMS WITH TWO VARIABLES AND THEIR APPLICATIONS

本文主要研究二元C∞函数芽环中函数芽的性质问题.利用Mather有限决定性定理和C∞函数的右等价关系，获得了带有任意4次至k次齐次多项式pi（x，y），qi（x，y）（i=4，5，…，k）的两类函数芽f1=x2y+∑i=4kpi（x，y），f2=xy2+∑i=4kqi（x，y）（k ≥ 5）的一个共同性质：若Mk2？M2J（fj）（j=1，2）且f1，f2的轨道切空间的余维分布均为ci=1（i=4，5，…，k-1），则对这里的i，pi（x，y）中xyi-1，yi的系数和qi（x，y）中xi-1y，xi的系数均为零.最后，利用该性质，给出了f1，f2和一类余维数为7的二元函数芽的标准形式.
In this paper, we mainly consider a property of function germs in the ring of C∞ functions germs of two variables. Using the Mather's theorem of finite determinacy and right equivalence of functions, a common property of two types of function germs f1=x2y+∑i=4kpi (x,y) and f2=xy2+∑i=4kqi (x,y)(k ≥ 5) with some arbitrary homogeneous polynomials pi (x,y) and qi(x, y)(i=4, 5, …, k) of degree from 4 to k is obtained. If Mk2 ？ M2J (fj)(j=1,2) and the codimension distribution of tangent space of orbits for f1,f2 are both ci=1(i=4,5,…,k-1), the coefficients of xyi-1 and yi in pi(x, y) are both zero, so are the coefficients of xi-1y and xi in qi(x, y). Finally, by this property, the normal forms of f1,f2 and a class of function germs of two variables with codimension 7 are given