论文标题
完全高度的$ t $ addic功能
Totally $T$-adic functions of small height
论文作者
论文摘要
令$ \ mathbb {f} _q（t）$为有限字段的一个变量中的有理函数字段。我们介绍了一个完全$ t $ - ad的函数的概念：一个超过$ \ mathbb {f} _q（t）$的代数函数，其最小的多项式在完成$ \ mathbb {f} _q（\！（\！（t）\！）$上完全折断。我们给出了两个证据，表明非构恒定的高度完全是$ t $ ad的函数，远离零，每个功能都提供了一个尖锐的下限。我们花费大部分论文提供了完全高度（通过算术动力学）和最小高度（通过几何和计算机搜索）的明确构造。我们还执行了一个大型计算机搜索，该搜索证明了某些类型的完全$ t $ ad-adic功能，其高度超过$ \ mathbb {f} _2（t）$不存在。在$ \ mathbb {f} _q（t）$上是否存在最低正高度的最低正高度的$ T $ ADIC功能是否存在无限的问题。最后，我们考虑了这些概念的类似物，这些概念是在额外的完整性假设下。
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.
