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Let $R=k[x_1,...,x_n]$ be the polynomial ring over an algebraically closed field $k$ and let $f,g\in R$. Assume that $f$ and $g$ are irreducible. How can I understand whether $k[x_1,...,x_n,\frac{1}{f}]$ and $k[x_1,...,x_n,\frac{1}{g}]$ are isomorphic?

Alternatively, consider the open sets $D(f)=\{p\in\mathbb{A}^n\mid f(p)\neq 0\}$ and $D(g)=\{p\in\mathbb{A}^n\mid g(p)\neq 0\}$. Is there an easy way to deduce whether $D(f)\cong D(g)$?

That amounts of checking $V(tf-1)\cong V(tg-1)$ in $k[x_1,...,x_n,t]$ and I believe that this should not be an easy problem.

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  • $\begingroup$ I think the only way this can happen is if there is an isomorphism $\phi:k[x_1,\ldots, x_n]$ to itself such that $\phi(f)=g$. $\endgroup$
    – Mohan
    Jan 12, 2019 at 1:43
  • $\begingroup$ Some one-variable examples: $k[x,1/x] \simeq k[x,1/x^2]$. But what about $k[x,1/f]$ and $k[x,1/g]$ when $f(x)=x(x-1)(x-2)$ and $g(x)=x(x-1)(x-3)$? $\endgroup$
    – user210229
    Jan 15, 2019 at 14:48

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