Rational functions sharing preimages and height functions

Abstract

Let $A$ and $B$ be non-constant rational functions over $\mathbb{C}$, and let $K \subset \mathbb{P}^1(\mathbb{C})$ be an infinite set. Using height functions, we prove that the inclusion $ A^{-1}(K) \subseteq B^{-1}(K) $ implies the inequality $ {\rm deg} B \geq {\rm deg} A $ in the following two cases: the set $K$ is contained in $\mathbb{P}^1(k)$, where $ k$ is a finitely generated subfield of $\mathbb{C}$, or the set $K$ is discrete in $\mathbb{C}$, and $A$ and $B$ are polynomials. In particular, this implies that for $A$, $B$, and $K$ as above, the equality $ A^{-1}(K) = B^{-1}(K) $ is impossible, unless $ {\rm deg} B = {\rm deg} A $.