The Special Nature Of Some Function Germs Of Two And Their Applications

In the study of the classification of C^∞real function germs with corank 2 and codimension 5 in [30], a special property of the function germs f₁ ( x, y) = x² y + p₁ ( x, y) with an arbitrary homogeneous polynomial p₁ ( x , y )of degree 4 was given.The orbit of f₁ ( x , y )is 4-open, which is equivalent to the fact that the coefficient of y⁴in p₁ ( x , y ) is non-zero and this property is applied to obtain normal forms of this class of germs.In this paper, the special property is generalized to some function germs with similar forms. As an application, their normal forms are given. Our generalizations contain two parts:Generalization 1 Let f₁ ( x , y )= x²y+ p₁ ( x , y), f₂ ( x , y ) = x³+ p₂( x , y),f₃ (x,y)=xy²+ p₃ ( x , y )and f₄ ( x,y)= y³+p₄ ( x , y )are function germs with arbitrary homogeneous polynomials p₁ ( x , y ), p₂ ( x , y ), p₃ ( x , y)and p₄ ( x , y )of degree 4, respectively, we have:Theorem1 (i) The orbit of f₁ ( x , y )( f₂( x , y )) is 4-open, which is equivalent to the fact that the coefficient of y⁴in p₁ ( x , y )( p₂ ( x , y ))is non-zero;(ii) The orbit of f₃( x , y )( f₄( x , y ))is 4-open, which is equivalent to the fact that the coefficient of x⁴in p₃ ( x , y )( p₄ ( x , y ))is non-zero. As an application of theorem1, we obtain:Theorem2 If f₁ ,f₂, f₃and f₄ have the property of theorem 1 , then f₁ , f₂,f₃ and f₄ are isomorphic to x² y±y⁴ , x³Â±y⁴ ,xy²Â±x⁴ and y³Â±x⁴, respectively. Further, the normal forms of f₁ and f₃are x² y±y⁴. The normal forms of f₂ and f₄ are x³Â±y⁴.Generalization 2 Based on the study of f₁ ,f₂, f₃ and f₄, we further consider the Where p_i , q_i ,u_i and v_i are all arbitrary homogeneous polynomials of degree i (i = 4,, k ; k≥5)and c_r^j = dim_R [( M^r (2) + M (2)·J ( f_j )) /( M^r+1(2) + M (2)·J ( f_j))]is the codimensional distribution of tangent spaces of f j( j = 5,6,7,8).Under the following codimensional distribution of tangent spaces of f j( j = 5,6,7,8), f₅ , f₆,f₇and f₈have the following special properties: orbits are k-open ,where k≥5.If the codimensional distribution of tangent space of f₅ is c_i⁵≡1, i = 4,5, , k-1,then the coefficients of xy^i-1 and yⁱ in p_i ( x,y)are zero. Similarly, if the codimensional distribution of tangent space of f₆ is c_i⁶≡1,i = 4,5, , k-1,then the coefficients of x^i-1 y and xⁱ in q_i ( x , y )are zero. Where p_i ( x,y) and q_i ( x , y ) are arbitrary homogeneous polynomials of degree i (i = 4, ,k ) with respect to x ,y . are k-open ,where k≥5.If the codimensional distribution of tangent space of f₇ is c_i⁷≡2, i = 4,5, , k-1,then the coefficients of x² y^i-2 ,xy^i-1 and yⁱ in u_i ( x , y )are zero. Similarly, if the codimensional distribution of tangent space of f 8 is c_i⁸≡2,i = 4,5, , k-1,then the coefficients of x^i-2 y² ,x^i-1 y and xⁱ in v_i ( x , y )are zero. Where u_i ( x , y )and v_i ( x , y )are arbitrary homogeneous polynomials of degree i (i = 4,5, , k ) with respect to x ,y .As an application of theorem 3, we obtain the following theorem 5 and corollary: they have the properties of theorem 3,then f 5 ,f 6are isomorphic to x² y±y^kand xy ²Â±x^k,respectively. Further, the normal forms of f₅ ,f₆ are x² y±y^k. Where p_i (x ,y) and q_i ( x , y ) are arbitrary homogeneous polynomials of degree i (i = 4, , k ) with respect to x ,y .Corollary Let f ( x , y )∈M³(2) is a function germ of two variables with codimension 7 and the codimensional distribution of tangent space of f is c₃ = c₄ = c₅ = 1,c₆ = 0, then the normal forms of f are x² y±y⁶.