On modules of integral elements over finitely generated domains
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Abstract:
This paper is motivated by the results and questions of Jason P. Bell and Kevin G. Hare in 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 $\mathcal {O}$ and 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 $\mathcal {O}$ and 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 Győry, 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 Győry 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. For (A), we show that except for some “degenerate” cases that can be explicitly described, there are at most $c_1$ such pairs $(m,n)$. This significantly strengthens some results of Bell and Hare. We also make some remarks on effectiveness and discuss further questions at the end of the paper.
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Additional Information
- Khoa D. Nguyen
- Affiliation: Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 886774
- Email: dknguyen@math.ubc.ca
- Received by editor(s): December 22, 2014
- Received by editor(s) in revised form: April 21, 2015
- Published electronically: August 18, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3047-3066
- MSC (2010): Primary 11D61; Secondary 11R99
- DOI: https://doi.org/10.1090/tran/6732
- MathSciNet review: 3605964