Implicit Mapping Theorem of Complex Variables and Riemann Surfaces

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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 m……