On intersections of fields of rational functions

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 $\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{X}\cdot \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{Y}$. 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.