Title: On intersections of fields of rational functions

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 $\deg X\cdot\deg Y$.
We give a complete characterization in the case where $X$ is a Galois covering,
and establish several related results.

Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=6745e950-7ffc-41cc-b19c-b44d0059494d