Implicit function theorem and uniformization (Q2779125)

Let \(F: B\times \Omega\rightarrow N\) be a family of holomorphic mapping where \(\Omega\) and \(N\) are complex holomorphic manifolds of the same dimension \(n\) and \(B\) is the unit ball in \({\mathbb C^k}\) as a parameter space. The author considers the case when the tangential mapping \(d_x (x\in \Omega)\) is not an isomorphism. Under a Frobenius type condition, he proves that the implicit function defined by \(F(t, x)=0\) for \(t\in B\) and \(x\in \Omega\) is ramified by power functions. A generalization of Puisseux series as well as an application is also discussed.