 Solution

Let $\zeta =e^{2\pi i/n}$ and define Möbius transformationsThey satisfyand the $2n$ maps $r^k$ and $s{r}^k$ are distinct. They therefore give a faithful action of the dihedral group $D_{2n}$ on the Riemann sphere.

The rational functionis invariant under both $r$ and $s$, so it is constant on every orbit. Conversely, for nonzero finite $z_1,z_2$, put $u=z_1^n$ and $v=z_2^n$. Equality of the function values givesIf $u=v$, then $z_2={\zeta }^k{z}_1=r^k{z}_1$ for some $k$. If $uv=1$, then $z_2={\zeta }^k/z_1=r^{k}s{z}_1$ for some $k$. Finally, zero and infinity both map to infinity and are interchanged by $s$. HenceThis is the orbit-separating invariant for the standard dihedral action on the Riemann sphere.