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Recall Serre's criterion for normality.

Let $f=f(t),g=g(t) \in \mathbb{C}[t]$, and assume that $m:=\deg(f) \geq 2$, $n:=\deg(g) \geq 2$. Denote $f=a_m t^m+\cdots +a_1t+a_0$ and $g=b_n t^n+\cdots +b_1t+b_0$.

I am interested in normality (= integral closedness) of the ring $R:= \mathbb{C}[f,g] \subset \mathbb{C}[t]$, which I further assume satisfies $\mathbb{C}(f,g)=\mathbb{C}(t)$.

How to apply Serre's criterion to such general $f$ and $g$? Is it possible to say something interesting about normality of $R$ in terms of $m,a_i,n,b_i$?

See this related question, which shows that if $m$ does not divide $n$ and $n$ does not divide $m$, then $R$ is not normal, but also $\mathbb{C}[t^3,t^6+t^2]$ is not normal though $3|6$.

See also this question concerning prime ideals in $\mathbb{C}[f,g]$.

Thank you very much!

user237522
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    For one dimensional domains, Serre criterion really says nothing. – Mohan Jun 17 '18 at 00:08
  • Thanks for your comment. What if we take $f,g,h \in \mathbb{C}[x,y]$, $R:=\mathbb{C}[f,g,h]$ with $\mathbb{C}(f,g,h)=\mathbb{C}(x,y)$? – user237522 Jun 17 '18 at 00:55
  • For example, $f=x$, and $(g,h)$ is a Jacobian pair. (It is known that $\mathbb{C}(x,g,h)=\mathbb{C}(x,y)$). Is Serre's criterion more helpful now? – user237522 Jun 17 '18 at 00:56

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