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.