Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11C08, 11R06, 11Y40

Retrieve articles in all journals with MSC (2000): 11C08, 11R06, 11Y40


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: http://dx.doi.org/10.1090/S0025-5718-06-01958-2
PII: S 0025-5718(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.