Totally $T$-adic functions of small height

Abstract

Let Fq(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 Fq(T ) and whose minimal polynomial splits completely over the completion Fq((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 F2(T ) do not exist. The problem of whether there exist infinitely many totally T -adic functions of minimum positive height over Fq(T ) remains open. Finally, we consider analogues of these notions under additional integrality hypotheses.