# Totally $T$-adic functions of small height

### Xander Faber

Institute for Defense Analyses, Bowie, USA### Clayton Petsche

Oregon State University, Corvallis, USA

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## 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.

## Cite this article

Xander Faber, Clayton Petsche, Totally $T$-adic functions of small height. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 4, pp. 699–732

DOI 10.4171/RLM/911