Some finiteness results on monogenic orders in positive characteristic

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This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let O is a finitely generated Z-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix s that is integral over O, describe all t such that O[s]=O[t]. (B) Fix s and t that are integral over O, describe all pairs (m,n)² such that O[sᵐ]=O[tⁿ]. 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 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 xᵖ. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this paper.