We have: $X$ a compact Riemann surface defined by $y^{2}=1-x^{6}$ and $P=(0,1) \in X$ a point given in local coordinates $(x,y)$. Furthermore, we have a meromorphic function $f(x,y)=y/x$ such that $f \in H^{0}(\mathcal{O}_{nP})$ for $n=3$. We know that our function is defined up to additive constant by its principal parts, i.e. $f(x)=a_{n}/x^{n}+\cdots+a_{1}/x+O(1)$ where the final term is something holomorphic. Given a holomorphic 1-form expressed near $P$ as $\omega(z)=(b_{0}+b_{1}z+\cdots+b_{n-1}z^{n-1}+O(z^{n}))dz$, we see that $\mathrm{Res}_{P}(f\omega)=b_{0}a_{1}+\cdots+b_{n-1}a_{n}$. That is, the vector $(b_{i})$ determines a linear constraint on the vector $(a_{i})$.

Using our function $f(x,y)$ I want to explicitly compute the principal parts $f(x)$ given above, and then conclude $\mathrm{Res}_{P}(f\omega)=0$ for all $\omega \in \Omega(X)$. I'm not quite sure how to do this in practice. Can someone demonstrate this to me? There seems to be a dearth of explicit examples of such a problem in the literature. Thanks very much.


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