Let be the field of rational functions in one variable over a finite field. We introduce the notion of a totally -adic function: one that is algebraic over and whose minimal polynomial splits completely over the completion . We give two proofs that the height of a nonconstant totally -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 -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 -adic functions of minimum height over do not exist. The problem of whether there exist infinitely many totally -adic functions of minimum positive height over remains open. Finally, we consider analogues of these notions under additional integrality hypotheses.
Cite this article
Xander Faber, Clayton Petsche, Totally -adic functions of small height. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 4, pp. 699–732DOI 10.4171/RLM/911