How to find the genus of curve $x^2-\cos(y)=c$?

Question

$x^2-\cos(y)=c$ is not an algebraic curve, and its genus may depend on the constant $c$, how to find them? In general, do we have any algorithms for transcendental curves?

Answer 1

This curve $X$ is the Riemann surface of the multivalued function $y \mapsto \sqrt{c + \cos(y)}$. The function $\cos(y) $ is surjective, hence there are infinitely many branch points, given by $(x,y)= (0,y_0)$ where $y_0 \in \mathbb C$ is a solution of $\cos(y_0) = -c$.

It follows that the genus of the curve is infinite, independently of $c$, because for each pair of branch points $(y_0,y_1)$ you get a new non-trivial cycle. If you take the $y_i$'s all disjoint they will be independent in homology.