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Transactions of the American Mathematical Society

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On modules of integral elements over finitely generated domains

Author: Khoa D. Nguyen
Journal: Trans. Amer. Math. Soc. 369 (2017), 3047-3066
MSC (2010): Primary 11D61; Secondary 11R99
Published electronically: August 18, 2016
MathSciNet review: 3605964
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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:

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]$.
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

Keywords: Unit equations over finitely generated domains, uniform bounds, effective methods
Received by editor(s): December 22, 2014
Received by editor(s) in revised form: April 21, 2015
Published electronically: August 18, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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