Consider the Riemann sphere $\hat{\mathbb{C}} := \mathbb{C} \cup \{ \infty \}$. One often considers polynomials on the Riemann sphere, i.e. maps
\begin{equation} P: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}, P(z) = z \mapsto \sum_{n=0}^N a_n z^n \end{equation}
and rational functions on the Riemann sphere, which are quotients of polynomials.
On smooth manifolds, as far as I would think, polynomials and rational functions cannot be invariantly characterized. But what about the Riemann sphere or Riemann surfaces in general?
I would think, that rational functions are invariantly characterized on the Riemann sphere, since conformal automorphisms of the Riemann sphere are rational maps (Möbius transformations) by themselves. Hence any Möbius transformation maps a rational function on a rational function. Hence, I would think that in any chart associated with the holomorphic structure rational functions stay rational functions. Is this true, or am I overlooking something?
If above is true, what is the case for general Riemann surfaces?
Thank you!