Some Finiteness Results on Monogenic Orders in Positive Characteristic. (24th December 2016) - Bell, Jason P; Nguyen, Khoa D
Abstract: 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)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 xx^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.