I am trying to do an exercise for a course about Riemann surfaces and for that, I need to find the valency of all points with respect to a function in other words, the branch points with their respective valencies. The book I am using is A Course in Complex Analysis and Riemann Surfaces by Wilhelm Schlag, but as far as I know this question does not appear in the book.
Valency is defined as follows:
Let $f:M\rightarrow N$ be an analytic and nonconstant map between Riemann surfaces. Then the valency of $f$ at a point $p\in M$, denoted by $\nu_f(p)$ is the unique positive integer $n$ with the property that in charts $(U,\phi)$ around $p$ with $\phi(p)=0$ and $(V,\psi)$ around $f(p)$ with $\psi(f(p))=0$ we have $(\psi\circ f\circ \phi^{-1})(z)=(zh(z)^n)$ where $h(0)\neq 0$.
Now to define $f$. Let $h$ be a polynomial of degree $N\geq 2$ with $N$ distinct complex zeros. Then $$S=\{(z,w)\in\mathbb{C}^2\mid w^2 = h(z)\}$$ is a smooth affine plane curve.
If $N$ is even, then $S$ can be made into a compact Riemann surface $M$ by adding two points at infinity (this was shown earlier).
Now we can define $f$. Let $f:M\rightarrow N = \mathbb{C}_\infty$ be the projection map $$p\in M \mapsto \begin{cases}z, \text{ if }p=(z,w)\in S,\\ \infty, \text{ if }p\text{ is one of the points at infinity,}\end{cases}$$ where $\mathbb{C}_\infty$ is the Riemann sphere.
Now I want to find the valencies of $f$ at each point $p\in M$. The only way I see to do this is to find explicit charts and calculated the first derivative of $\psi\circ f\circ\phi^{-1}$, but I do not think this is the way to go. I have seen that for zeros of $h$ the valency is $2$ and for other points $1$. But I have no idea why that is.