Some finiteness results on monogenic orders in positive characteristic

Abstract

This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\mathcal{O}$ is a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix $s$ that is integral over $\mathcal{O}$, describe all $t$ such that $\mathcal{O}[s]=\mathcal{O}[t]$. (B) Fix $s$ and $t$ that are integral over $\mathcal{O}$, describe all pairs $(m,n)\in\mathbb{N}^2$ such that $\mathcal{O}[s^m]=\mathcal{O}[t^n]$. In this paper, we solve these problems and provide a uniform bound for a certain "discriminant form equation" that is closely related to Problem (A) when $\mathcal{O}$ has characteristic $p>0$. While our general strategy roughly follows [EG85] and [Ngu15], many new delicate issues arise due to the presence of the Frobenius automorphisms $x\mapsto x^p$. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this paper.