Rationality of meromorphic functions between real algebraic sets in the plane

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 Poincaré(1907), Tanaka(1962 and Alexander(1974)'s rationality results for 2m−1 dimensional sphere in C m when m ≥ 2. 2010 Mathematics Subject Classification. 30C99, 30D99.