Totally $T$-adic functions of small height

IF 0.6 4区 数学 Q3 MATHEMATICS
X. Faber, Clayton Petsche
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引用次数: 0

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.
总$T$-小高度的adic函数
让$\mathbb{F}_q(T) $是有限域上一个变量中有理函数的域。我们引入了一个完全$T$adic函数的概念:一个在$\mathbb上代数的函数{F}_q(T) $,并且其最小多项式在完备$\mathbb上完全分裂{F}_q(\!(T)\!)$。我们给出了两个证明,证明了一个非常的全$T$adic函数的高度离零有界,每一个都提供了一个尖锐的下界。我们在论文的大部分时间里提供了小高度(通过算术动力学)和最小高度(通过几何和计算机搜索)的总$T$-dic函数的显式构造。我们还执行了一个大型计算机搜索,证明了$\mathbb上最小高度的某些类型的完全$T$-adic函数{F}_2(T) $不存在。$\mathbb上是否存在无穷多个最小正高度的完全$T$-adic函数的问题{F}_q(T) $仍然开放。最后,我们在附加的完整性假设下考虑这些概念的类似物。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Rendiconti Lincei-Matematica e Applicazioni
Rendiconti Lincei-Matematica e Applicazioni MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.30
自引率
0.00%
发文量
27
审稿时长
>12 weeks
期刊介绍: The journal is dedicated to the publication of high-quality peer-reviewed surveys, research papers and preliminary announcements of important results from all fields of mathematics and its applications.
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