Some finiteness results on monogenic orders in positive characteristic

TL;DR

This paper investigates finiteness properties of monogenic orders in positive characteristic, extending previous results from characteristic zero and addressing new challenges posed by Frobenius automorphisms.

Contribution

It solves key problems about monogenic orders in positive characteristic and establishes uniform bounds for related discriminant equations, adapting methods to handle Frobenius automorphisms.

Findings

Solved problems describing all $t$ with $ ext{O}[s]= ext{O}[t]$ in positive characteristic.

Provided uniform bounds for discriminant form equations in positive characteristic.

Extended classical results to include the effects of Frobenius automorphisms.

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.