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 $\mathrm{deg}B\ge \mathrm{deg}A$ 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 $\mathrm{deg}B=\mathrm{deg}A$.