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arXiv · 1403.7480

Comments on the height reducing property II

Shigeki Akiyama·Jörg M. Thuswaldner·Toufik Zaïmi
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Abstract

A complex number $\alpha$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[\alpha]=F[\alpha]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $\alpha$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_{\alpha}(0)\right\vert \},$ where $M_{\alpha}$ is the minimal polynomial of $\alpha$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $\alpha$. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.

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Shigeki Akiyama, Jörg M. Thuswaldner, Toufik Zaïmi. 2014-03-28. Comments on the height reducing property II. https://arxiv.org/abs/1403.7480

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