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THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

Part of: Polynomials and matrices Computational number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  17 February 2011

YANHUA LIANG  and
QIANG WU
YANHUA LIANG
Affiliation:
Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, PR China (email: happy08@swu.edu.cn)
QIANG WU*
Affiliation:
Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, PR China (email: qiangwu@swu.edu.cn)
*
✉For correspondence; e-mail: qiangwu@swu.edu.cn
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Abstract

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Let α be a totally positive algebraic integer of degree d≥2 and α1=α,α2,…,αd be all its conjugates. We use explicit auxiliary functions to improve the known lower bounds of Sk/d, where Sk=∑ di=1αki and k=1,2,3. These improvements have consequences for the search of Salem numbers with negative traces.

Keywords

absolute tracetotally positive algebraic integerSalem numberexplicit auxiliary functionsemi-infinite linear programmingLLL algorithm

MSC classification

Secondary: 11C08: Polynomials 11Y40: Algebraic number theory computations 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 3 , June 2011 , pp. 341 - 354
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The corresponding author was supported by the Natural Science Foundation of Chongqing grant CSTC no. 2008BB0261.

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