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Totally $T$-adic functions of small height
- Source :
- Rendiconti Lincei - Matematica e Applicazioni. 31:699-732
- Publication Year :
- 2021
- Publisher :
- European Mathematical Society - EMS - Publishing House GmbH, 2021.
-
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 $\mathbb{F}_2(T)$ 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.<br />25 pages; source code for computations in the paper available at https://github.com/RationalPoint/T-adic
Details
- ISSN :
- 11206330
- Volume :
- 31
- Database :
- OpenAIRE
- Journal :
- Rendiconti Lincei - Matematica e Applicazioni
- Accession number :
- edsair.doi.dedup.....99ffa0fa991d321c8f3fff8026e5751d