JournalsrlmVol. 31, No. 4pp. 699–732

Totally TT-adic functions of small height

  • Xander Faber

    Institute for Defense Analyses, Bowie, USA
  • Clayton Petsche

    Oregon State University, Corvallis, USA
Totally $T$-adic functions of small height cover
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Let Fq(T)\mathbb F_q (T) be the field of rational functions in one variable over a finite field. We introduce the notion of a totally TT-adic function: one that is algebraic over Fq(T)\mathbb F_q (T) and whose minimal polynomial splits completely over the completion Fq((T))\mathbb F_q ((T)). We give two proofs that the height of a nonconstant totally TT-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 TT-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 TT-adic functions of minimum height over Fq(T)\mathbb F_q (T) do not exist. The problem of whether there exist infinitely many totally TT-adic functions of minimum positive height over Fq(T)\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 TT-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