
> 
> 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 $n$?**



Is $\mathbb{C}[x^{\pm 1},y]/\langle y^3-(x^2+ax+b) \rangle$ a $n$-point ring?
No (assuming that $a^2 \neq 4b$, so the quadratic is not a square). In fact, the fields $\mathrm{Frac}(R)$ and $\mathbb{C}(t)$ are not isomorphic. From a sophisticated perspective, the point is that the former has genus one and the latter has genus $0$. In general, this is a hard notion to define in an elementary way; see the discussion [here](https://mathoverflow.net/questions/152/how-do-you-see-the-genus-of-a-curve-just-looking-at-its-function-field).


Let's do something elementary and check that $\mathbb{C}[x,y]/(y^2-x^3-1)$ is not an $n$-point ring. If it were, we could identify it with a subring of $\mathbb{C}(t)$, let $x$ correspond to $p(t)/q(t)$ and $y$ to $r(t)/s(t)$ in lowest terms. Comparing poles, we must have $q(t) = d(t)^3$ and $s(t) = d(t)^2$ for some polynomial $d$, so $(p(t)/d(t)^2)^3 = (r(t)/d(t)^3)^2+1$ or $p(t)^3=r(t)^2+s(t)^6$. This would violate the [Mason-Stothers theorem](https://en.wikipedia.org/wiki/Mason%E2%80%93Stothers_theorem), and you can mimic any proof of that theorem to see that this is impossible.


