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 nonsingular 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?

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?

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 Riemann surface like that $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$.

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 allowed 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:

Proposition 1: Let $p(t)=\sum_{i\in\mathbb{Z}}a_it^i\in\mathbb{C}[t]$ 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?

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 nonsingular 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?

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 allowed poles in a Riemann surface like that $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$.

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 allowed 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:

Proposition 1: Let $p(t)=\sum_{i\in\mathbb{Z}}a_it^i\in\mathbb{C}[t]$ 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?

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 Riemann surface like that $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$.

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 allowed 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:

Proposition 1: Let $p(t)=\sum_{i\in\mathbb{Z}}a_it^i\in\mathbb{C}[t]$ 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?