隐函数定理与单值化

令 F:B×Ω→ N是一族全纯映射 ,其中Ω与 N为同维数 n的复流形 ,B∈ Ck是单位圆盘作为参数空间 .隐函数定理考查 F- 1(0 )之构造 .本文中我们考虑切映射 dz F(x∈Ω)不是线性同构的情况 ,并在一个 Frobenius型条件下证明了F(t,x) =0 ,　　 t∈ B,x∈Ω所定义的隐函数可以用幂函数单值化 .作为它的应用 ,我们给出了 Puisseux级数的一个推广 .

IMPLICIT FUNCTION THEOREM AND UNIFORMIZATION

Let F:B × Ω → N be a family of holomorphic mapping where Ω and N are complex holomorphic manifolds of the same dimension n,B ∈ Ck is the unit disc as parameter space.The implicit function function theorem considers the structure of F-1(0) . In this paper, we consider the case when the tangential mapping dF(x ∈ Ω) is not an isomorphism. Under a Frobenius-type condition,we prove that the implicit function defined by F(t,x) = 0, t ∈ B, x ∈- Ωin ramified by power functions. A gereralization of Puisseux series is also given as an application.