On the smallest value of the maximal modulus of an algebraic integer
Authors:
Georges Rhin and Qiang Wu
Journal:
Math. Comp. 76 (2007), 10251038
MSC (2000):
Primary 11C08, 11R06, 11Y40
Published electronically:
December 29, 2006
MathSciNet review:
2291848
Fulltext PDF Free Access
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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 SchinzelZassenhaus 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.univmetz.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/S0025571806019582
PII:
S 00255718(06)019582
Keywords:
Algebraic integer,
maximal modulus,
SchinzelZassenhaus 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.
