Some Finiteness Results on Monogenic Orders in Positive Characteristic

This work is motivated by the articles [9] and [19] in which the following two problems are solved. Let O be 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) is an element of N-2 such that O[s(m)] = O[t(n)]. In this article, 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 [9] and [19], many new delicate issues arise due to the presence of the Frobenius automorphism x (bar right arrow) 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 article.