Definition: Let $\{a_1,\dots,a_n\}$ be any $n$ distinct points on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ with coordinate $s$, and let $R$ be the ring of rational functions with poles allowed only in $\{a_1,\dots,a_n\}$. This ring is called $n$-point ring.

It is well known that

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-t-a^2 \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+1}] $ is a $3$-point ring, and

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-(t^2-2bt+1) \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+b+1},\frac{1}{t+b-1}] $ is a $4$-point ring.

It can be proven just using some isomorphisms.

Is $R=\mathbb{C}[x^{\pm 1},y]/\langle y^3=x^2+ax+b \rangle$ a $n$-point ring for some $n$?

Definition: Let $\{a_1,\dots,a_n\}$ be any $n$ distinct points on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ with coordinate $s$, and let $R$ be the ring of rational functions with poles allowed only in $\{a_1,\dots,a_n\}$. This ring is called $n$-point ring.

It is well known that

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-t-a^2 \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+1}] $ is a $3$-point ring, and

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-(t^2-2bt+1) \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+b+1},\frac{1}{t+b-1}] $ is a $4$-point ring.

It can be proven just using some isomorphisms.

Is $R=\mathbb{C}[x^{\pm 1},y]/\langle y^3-(x^2+ax+b) \rangle$ a $n$-point ring for some $n$?

Definition: Let $\{a_1,\dots,a_n\}$ be any $n$ distinct points on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ with coordinate $s$, and let $R$ be the ring of rational functions with poles allowed only in $\{a_1,\dots,a_n\}$. This ring is called $n$-point ring.

It is well known that

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-t-a^2 \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+1}] $ is a $3$-point ring, and

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-(t^2-2bt+1) \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+b+1},\frac{1}{t+b-1}] $ is a $4$-point ring.

It can be proven just using some isomorphisms.

Does $R=\mathbb{C}[x^{\pm 1},y]/\langle y^3=x^2+ax+b \rangle$ is a $n$-point ring for some $n$?

Definition: Let $\{a_1,\dots,a_n\}$ be any $n$ distinct points on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ with coordinate $s$, and let $R$ be the ring of rational functions with poles allowed only in $\{a_1,\dots,a_n\}$. This ring is called $n$-point ring.

It is well known that

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-t-a^2 \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+1}] $ is a $3$-point ring, and

• $\mathbb{C}[t^{\pm 1},u]/\langle u^2-(t^2-2bt+1) \rangle \cong \mathbb{C}[t^{\pm 1},\frac{1}{t+b+1},\frac{1}{t+b-1}] $ is a $4$-point ring.

It can be proven just using some isomorphisms.

Is $R=\mathbb{C}[x^{\pm 1},y]/\langle y^3=x^2+ax+b \rangle$ a $n$-point ring for some $n$?