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 $\mathcal{O}$ be 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 article, 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 [9 ] and [19 ], many new delicate issues arise due to the presence of the Frobenius automorphism $x\mapsto x^p$ . Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this article.