Rationality of meromorphic functions between real algebraic sets in the plane

TL;DR

This paper investigates conditions under which meromorphic functions between real algebraic sets in the plane are necessarily rational, extending classical results about rationality on spheres to planar algebraic sets.

Contribution

It demonstrates that certain meromorphic functions mapping specific real algebraic sets in the plane must be rational, using Schwarz reflection functions, including an analog of classical sphere rationality results.

Findings

Meromorphic functions on the unit circle are rational under certain conditions.

Extension of classical sphere rationality results to planar algebraic sets.

Use of Schwarz reflection functions to establish rationality.

Abstract

We study one variable meromorphic functions mapping a planar real algebraic set $A$ to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain $A$, these meromorphic functions must be rational. In particular, when $A$ is the standard unit circle, we obtain an one dimensional analog of Poincar\'e(1907), Tanaka(1962) and Alexander(1974)'s rationality results for $2m-1$ dimensional sphere in $\mathbb{C}^m$ when $m\ge 2$.