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Suppose that we have the following curve $X$ on $\mathbb C^2$ given by the equation $$ x^3y+y^3+x=0 $$ (The particular example is not important - I have seen several examples with the same confusion.) The first thing I am wondering is that how I can prove it is a Riemann surface with charts given by coordinate projections. In other words, how I could show that $x\mapsto y$ is biholomorphic locally. But I just could not figure out a way to do that without writing done the formula of this implicit function directly. Apparently I need some implicit function theorem for complex functions - but I am learning about Riemann surfaces, so there won't be such a theorem in my book (since everything is one dimensional), so should there be a way to avoid such an theorem?

As we get to higher powers of $x,y$, an explicit formula is impossible. How can I justify that the map $x\mapsto y$ is biholomorphic in general?

The second question is somewhat related, is it possible for me to count the number of ramification and branch points of the coordinate projection maps $\pi_x$, given that I am not able to come up with a formula for them directly?

Ma Joad
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    the inverse function theorem (and hence implicit function theorem, and hence constant rank theorem) holds just as well in the complex setting. It holds very generally in the Banach space setting (over $\Bbb{R}$ or $\Bbb{C}$), see here for some remarks. In fact essentially the same Banach’s contraction-mapping fixed point theorem based proof goes through regardless of the underlying field. – peek-a-boo Jun 04 '23 at 20:02

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