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?**

The ring R is the coordinate ring of a curve $\Sigma$ of genus $g$ with $n$ points removed.Then he simply claims the Proposition 1. $\endgroup$ – Felipe Dec 28 '18 at 20:51