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For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ratio of two polynomials.)

For example, if $D$ is a proper convex cone, then we can choose $h$ to be just a linear diffeomorphism from $D$ to $\mathbb{R}^2_+$. What other domains $D$ are rationally diffeomorphic to $\mathbb{R}^2_+$?

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    $\begingroup$ Aren't the only rational functions whose inverse is also rational, the projective linear ones? $\endgroup$
    – Yaakov Baruch
    Commented Oct 19, 2015 at 7:43
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    $\begingroup$ @YaakovBaruch In dimension one an invertible rational map must be projective linear, but not necessarily in dimension two and higher. For example $(x,y)\mapsto (x,y+f(x))$ is invertible for any rational function $f$. $\endgroup$
    – Julian Rosen
    Commented Oct 19, 2015 at 19:01

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