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On the smallest value of the maximal modulus of an algebraic integer


Authors: Georges Rhin and Qiang Wu
Journal: Math. Comp. 76 (2007), 1025-1038
MSC (2000): Primary 11C08, 11R06, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-06-01958-2
Published electronically: December 29, 2006
MathSciNet review: 2291848
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Abstract: The house of an algebraic integer of degree $ d$ is the largest modulus of its conjugates. For $ d\leq 28$, we compute the smallest house $ >1$ of degree $ d$, say m$ (d)$. As a consequence we improve Matveev's theorem on the lower bound of m$ (d).$ We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer $ \boldsymbol \alpha $ whose house is equal to m$ (d)$ is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in $ \mathbb{C}.$ They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer $ \boldsymbol \alpha $ whose house is small.


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Additional Information

Georges Rhin
Affiliation: UMR CNRS 7122, Département UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
Email: rhin@math.univ-metz.fr

Qiang Wu
Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, China
Email: qiangwu@swu.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-06-01958-2
Keywords: Algebraic integer, maximal modulus, Schinzel-Zassenhaus conjecture, Mahler measure, Smyth's theorem, Perron numbers, explicit auxiliary functions, integer transfinite diameter.
Received by editor(s): December 24, 2005
Received by editor(s) in revised form: December 28, 2005
Published electronically: December 29, 2006
Additional Notes: Qiang Wu was supported in part by the Natural Science Foundation of Chongqing grant CSTC no. 2005BB8024
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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