Working through some problems in Rick Miranda's Algebraic Curves and Riemann Surfaces and wanna make sure I understand things correctly.
Problem V.1.A states: Let $X$ be the hyperelliptic curve defined by $y^2 = x^5 - x$. Note that $x$ and $y$ are meromorphic functions on $X$. Compute the principal divisors div$(x)$ and div$(y)$.
My reasoning: the function $x$, which is just projection onto the first coordinate, doesn't have any poles as $\infty$ is not included in our space at all. The only zero of $x$ is the point $(0,0)$ (as $x=0$ immediately gives $y=0$). I think that $(0,0)$ has order 2, because every other point $x_0$ has 2 points in its preimage (corresponding to the two roots of $x_0$) for $x_0\neq 0$. So div$(x) = 2\cdot (0,0)$.
Next I must address the function $y$, which is just projection onto the second coordinate.
Anyone care to comment/correct?