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.

中文翻译：

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小高度的完全$ T $ -adic函数

设$ \ mathbb F_q（T）$为有限域中一个变量的有理函数域。我们介绍一个完全$ T $ -adic函数的概念：一个在$ \ mathbb F_q（T）$上代数的函数，并且其最小多项式在完成$ \ mathbb F_q（（T））$上完全分裂。我们给出两个证明，一个非恒定的完全$ T $ -adic函数的高度与零有界，每一个都提供了一个尖锐的下界。我们花费大量的论文来提供完全$ T $ -adic函数的显式构造，该函数具有小高度（通过算术动力学）和最小高度（通过几何和计算机搜索）。我们还执行了大型计算机搜索，证明最小高度超过$ \ mathbb F_q（T）$的某些类型的完全$ T $ -adic函数不存在。关于在$ \ mathbb F_q（T）$之上的最小正高度是否存在无限多个完全$ T $ -adic函数的问题仍然存在。最后，我们在其他完整性假设下考虑这些概念的类似物。