On modules of integral elements over finitely generated domains

This paper is motivated by the results and questions of Jason P. Bell and Kevin G. Hare in the paper "On $\mathbb{Z}$-modules of algebraic integers" (Canad. J. Math. Vol. 61, 2009). Let $\mathcal{O}$ be a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero. We investigate the following two problems: (A) Fix $q$ and $r$ that are integral over $\mathcal{O}$, describe all pairs $(m,n)\in\mathbb{N}^2$ such that $\mathcal{O}[q^m]=\mathcal{O}[r^n]$. (B) Fix $r$ that is integral over $\mathcal{O}$, describe all $q$ such that $\mathcal{O}[q]=\mathcal{O}[r]$. In this paper, we solve Problem (A), present a solution of Problem (B) by Evertse and Gy\H{o}ry, and explain their relation to the paper of Bell and Hare. In the following, $c_1$ and $c_2$ are effectively computable constants with a very mild dependence on $\mathcal{O}$, $q$, and $r$. For (B), Evertse and Gy\H{o}ry show that there are $N\leq c_2$ elements $s_1,\ldots,s_N$ such that $\mathcal{O}[s_i]=\mathcal{O}[r]$ for every $i$, and for every $q$ such that $\mathcal{O}[q]=\mathcal{O}[r]$, we have $q-us_i\in\mathcal{O}$ for some $1\leq i\leq N$ and $u\in\mathcal{O}^*$. This immediately answers two questions about Pisot numbers by Bell and Hare in ibid. For (A), we show that except some "degenerate" cases that can be explicitly described, there are at most $c_1$ such pairs $(m,n)$. This significantly strengthens some results in ibid. We also make some remarks on effectiveness and discuss further questions at the end of the paper.