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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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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

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