There are two types of Belyi’s Theorems for curves defined over finite fields of characteristic [Formula: see text], namely the Wild and the Tame [Formula: see text]-Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi’s theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame [Formula: see text]-Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi’s Theorems unlike the known ones that use technical results from Algebraic Geometry.

中文翻译：

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Belyi 的正特征定理

定义在特征[公式：见正文]的有限域上的曲线有两种类型的Belyi定理，即Wild和Tame [公式：见正文]-Belyi定理。在本文中，我们用函数域的语言来讨论它们。特别是，我们为 [Y. Sugiyama 和 S. Yasuda，特征二中的 Belyi 定理，Compos。数学。156(2) (2020) 325–339]，这导致了 Tame [公式：见正文]-Belyi 定理的独立证明。此外，我们为 Belyi 定理提供了统一和简单的证明，这与使用代数几何技术结果的已知证明不同。