Totally $T$-adic functions of small height

Abstract

Let ${\mathbb{F}}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce the notion of a totally $T$-adic function: one that is algebraic over ${\mathbb{F}}_q(T)$ and whose minimal polynomial splits completely over the completion ${\mathbb{F}}_q((T))$. We give two proofs that the height of a nonconstant totally $T$-adic function is bounded away from zero, each of which provides a sharp lower bound. We spend the majority of the paper providing explicit constructions of totally $T$-adic functions of small height (via arithmetic dynamics) and minimum height (via geometry and computer search). We also execute a large computer search that proves certain kinds of totally $T$-adic functions of minimum height over ${\mathbb{F}}_q(T)$ do not exist. The problem of whether there exist infinitely many totally $T$-adic functions of minimum positive height over ${\mathbb{F}}_q(T)$ remains open. Finally, we consider analogues of these notions under additional integrality hypotheses.