ExamplesLet \(P(z,w)\) be a polynomial of two complex variables, i.e. \(P\) maps \(\mathbb{C} \times \mathbb{C}\) into \(\mathbb{C}\). We also want an inverse mapping theorem, just like what we have in multivariable calculus. For example, we may consider polynomials like \[P(z,w) = w^2-1-z^3.\] We may get a function determining \(w\) by \(z\) like \[w =\sqrt{1+z^3}\] when \(P=0\). This looks pretty good but it makes no sense. The reason is simple: we don't feel like square roots in complex varia...