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

Published electronically:
December 29, 2006

MathSciNet review:
2291848

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Abstract | References | Similar Articles | Additional Information

Abstract: The house of an algebraic integer of degree is the largest modulus of its conjugates. For , we compute the smallest house of degree , say m. As a consequence we improve Matveev's theorem on the lower bound of m We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer whose house is equal to m 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 They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer 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.