两类二元函数芽的一个共同性质和其亚标准形式

A Common Property of Two Types of Function Germs with Two Variables and Their Mild Normal Forms

利用J.N.Mather有限决定性定理和光滑函数芽的右等价关系,给出了带有任意4次至k次齐次多项式p_i(x,y),q_i(x,y)(i=4,5,…,k)的两类二元函数芽f_i=x^3+∑_(i=4)~kp_i(x,y),f_2=y^3+∑_(i=4)~k=4q_i(x,y)(k≥5)的一个共同性质:若M_2~kM_2J(f_j)(j=1,2)且f_1,f_2的轨道切空间的余维分布均为c_i=2(i=4,5,…,k-1),则对这个i,p_i(x,y)中x^2y^(i-2),xy^(i-1),y^i的系数和q_i(x,y)中x^(i-2)y^2,x^(i-1)y,x^i的系数均为零.最后,利用该性质,给出了f_1,f_2和一类余维数为8的二元函数芽的亚标准形式.

In this paper, Using the J. N. Mather＇s theorem of finite determinacy and right equivalence of functions in the theory of singularities, a common property of two types of function germs f1= x^3＋Σ（i=4）~k pi（s,y） and f2= y^3＋Σ（i=4）~k qi（s,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 given.If M2~kM2J（fj）（j = 1,2） and the codimension distribution of tangent space of orbits for f1,f2 are both ci= 2（i = 4,5,…,k-1）, the coefficients of x^2y^i-2,xy^i-1 and y^i in pi（x,y） are all zero, so are the coefficients of x^i-2y^2,x^（i-1）y and x^i 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 8 are given.