I have a question on the exact definition of branch points (on Riemann surfaces) I have the following definition::
Let $f:X \rightarrow Y$ be a holomorphic function between Riemann surfaces. $x \in X$ is a branch point, if the multiplicity, with which $f$ takes the value $f(x)$ in $x$ is $>1$. The mmultiplicity can be computed by expressing $f$ as a function between open susets of $\mathbb{C}$ by using charts.
In other words, branch points are the points where $f$ has no neighborhood on which it is injective.
For example, look at the function
$f: \mathbb{P}^1 \rightarrow \mathbb{P}^1, z \mapsto z^2 + \dfrac{1}{z^2}$
I computed the branch points to be $1, 0, \infty$ by constructing the function with the charts.
Now, I found this exercise on the internet, where they computed the points $\pm 2$ and $\infty$ as branch points.
Did I make a mistake or is there another definition for branch points?
Note that $f(-z)=f(z), f(1/z)=f(z)$ so certainly if $z_0$ is a branch point $\pm 1/z_0, -z_0$ are other branch points.
Also $f(-z)=f(z)$ implies that $0$ is a branch point, as around $0$ $f(-a)=f(a)$. Hence $0, \infty $ are branch points.
For seeking others, one can make stupid computations $f'(z)=2z-{2\over z^3}$, and $f'(z)=0$ iff $z^4=1$, ie $z\in \{1,-1,i,-i\}$. Conclusion 6 branch points$\{0, \infty,1,-1,i,-i\}$.