On intersections of fields of rational functions

TL;DR

This paper investigates the conditions under which the intersection of fields generated by rational functions has a finite degree extension, providing a complete characterization for Galois coverings and exploring related functional equations and Riemann surface maps.

Contribution

It offers a complete characterization of the intersection field extension for Galois coverings and studies related functional equations and Riemann surface mappings.

Findings

Characterization of when the field extension is finite for Galois coverings.

Results on the functional equation involving rational functions and Galois coverings.

Analysis of analogous problems for holomorphic maps between Riemann surfaces.

Abstract

Let $X$ and $Y$ be rational functions of degree at least two with complex coefficients such that $\mathbb{C}(X,Y)=\mathbb{C}(z)$. We study the problem of determining when the field extension $[\mathbb{C}(z):\mathbb{C}(X)\cap\mathbb{C}(Y)]$ is finite and attains the minimal possible degree ${\rm deg X}\cdot{\rm deg Y}$. We give a complete characterization in the case where $X$ is a Galois covering. We also establish several related results concerning the functional equation $A \circ X = Y \circ B$ in rational functions, in the case where one of the functions involved is a Galois covering. Finally, we consider an analogous problem for holomorphic maps between compact Riemann surfaces.