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

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- by Georges Rhin and Qiang Wu;
- Math. Comp.
**76**(2007), 1025-1038 - DOI: https://doi.org/10.1090/S0025-5718-06-01958-2
- Published electronically: December 29, 2006
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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 $\text {m}(d)$. As a consequence we improve Matveev’s theorem on the lower bound of $\text {m}(d).$ 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 $\text {m}(d)$ 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.## References

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## Bibliographic 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
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 1025-1038 - MSC (2000): Primary 11C08, 11R06, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-06-01958-2
- MathSciNet review: 2291848