Counting the number of poles for rational functions in a coordinate ring of a curve

Question

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in a coordinate ring $R$ of a curve $\Sigma$ with $R=\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$.

I'm doing an extensive research in previous references looking for a formula that could give to me that number and I kind of feel that it is just a language problem between algebraic-geometry and the papers I'm dealing with. Am I Right?

Sometimes in my research, the calculation of poles showed up like a simple statement that makes me feel that perhaps it could be a basic fact in algebraic geometry. I'll give an example simply copy pasted from one of papers I'm dealing with. Let R the coordinate ring of a curve $Σ$ of genus $g$ with $n$ points removed. Then he simply claims the following proposition.

Proposition 1: Considering $p(t)=\sum_{i\in\mathbb{Z}}a_it^i\in\mathbb{C}[t]$ and $R=\mathbb{C}[t,t^{-1},u]/\langle u^2-p(t)\rangle$.

The number $n$ where poles are allowed in $R$ depends on $p(t)$ according to the formula $n=4-r$ where $r$ is the number of ramified points in $\{0,\infty \}$: $0$ is ramified exactly when the constant term $a_0=0$, and $\infty$ is ramified exactly when the degree $d$ is odd.

Even opening all the references I cound't find some talking explicit about how to reach the Proposition 1 (that I call the hyperelliptic case) . My research colleges working in algebra also doesn't know how to find it.

I would like to know how to count the allowed poles in $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$ (that I call the superelliptic case). Is there any formula?

Answer 1

I think the best interpretation of the paper you link to is that they mean a Riemann surface is a nonsingular projective algebraic curve and they have simply forget to include the condition that $p(t)$ should be squarefree. For instance in Theorem 2.1 they use the algebraic de Rham theorem, which as stated is only right for smooth varieties, and state that the topological Euler characteristic is $2-2g$.

Thus I think the most reasonable interpretation is to mandate that $\operatorname{Spec} R$ be nonsingular, i.e. that $p(t)$ has all zeroes of order one, except possibly $t=0$, and ask what are the number of points needed to form a projective nonsingular compactification. Having done this, $R$ will the ring of meromorphic functions on a Riemann surface (in the usual sense) with $n$ points removed.

In this case, we can take $n = \gcd(i_0, m) + \gcd(i_\infty,m)$ where $i_0$ is the lowest $i$ such that $a_i \neq 0$ and $i_\infty$ is the greatest $i$ such that $a_i$ is not zero.

The proof can be done in many ways. One is topological. If we look at the covering defined by the equation $u^m =p(t)$ of a punctured disc around zero, we will see that each connected component is a punctured disc and requires one extra point to be filled, so the number of extra fillings is the number of connected components. Similarly, we will see that the number of points in a fiber is $m$, and that traveling in a loop around the disc connects the $j$th point to the $j+i_0$th point, so the number of connected components is the number of orbits of translation by $i_0$ on $\mathbb Z/m$, which is $\gcd(m,i_0)$. The same is true at $i_{\infty}$. There are also algebraic, algebro-geometric, etc. formulations of this argument.