Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T09:15:17.616Z Has data issue: false hasContentIssue false
  • English
  • Français

THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

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

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: [email protected])
QIANG WU*
Affiliation:
Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

[1]Aguirre, J. and Peral, J. C., ‘The trace problem for totally positive algebraic integers’, in: Number Theory and Polynomials, London Mathematical Society Lecture Note Series, 352 (eds. McKee, J. and Smyth, C.) (Cambridge University Press, Cambridge, 2008), pp. 11–19, With an appendix by Jean-Pierre Serre.Google Scholar
[2]Anderson, E. J. and Nash, P., Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, Series in Discrete Mathematics and Optimization (Wiley, Chichester, 1987).Google Scholar
[3]Batut, C., Belabas, K., Bernardi, D., Cohen, H. and Olivier, M., GP-Pari version 2.2.12 (2000). Available at http://pari.math.u-bordeaus.fr.Google Scholar
[4]Borwein, P., Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics, 10 (Springer, New York, 2002).CrossRefGoogle Scholar
[5]Flammang, V., ‘Trace of totally positive algebraic integers and integer transfinite diameter’, Math. Comp. 78 (2009), 1119–1125.CrossRefGoogle Scholar
[6]Flammang, V., Grandcolas, M. and Rhin, G., ‘Small Salem numbers’, in: Number Theory in Progress (Zakopane-Koscielisko, 1997), Vol. 1 (de Gruyter, Berlin, 1999), pp. 165–168.CrossRefGoogle Scholar
[7]Flammang, V., Rhin, G. and Wu, Q., ‘The totally real algebraic integers with diameter less than 4’, Moscow J. Combin. Number Theory, to appear.Google Scholar
[8]Flammang, V., Rhin, G. and Sac-Épée, J. M., ‘Integer transfinite diameter and polynomials of small Mahler measure’, Math. Comp. 75 (2006), 1527–1540.CrossRefGoogle Scholar
[9]Habsieger, L. and Salvy, B., ‘On integer Chebyshev polynomials’, Math. Comp. 66 (1997), 763–770.CrossRefGoogle Scholar
[10]McKee, J., ‘Computing totally positive algebraic integers of small trace’, Math. Comp. 80 (2011), 1041–1052.CrossRefGoogle Scholar
[11]McKee, J. and Smyth, C. J., ‘Salem numbers of trace −2 and trace of totally positive algebraic integers’, in: Algorithmic Number Theory. Proceedings of 6th Algorithmic Number Theory Symposium (University of Vermont 13–18 June 2004), Lecture Notes in Computer Science, 3076 (Springer, Berlin, 2004), pp. 327–337.Google Scholar
[12]McKee, J. and Smyth, C. J., ‘There are Salem numbers of every trace’, Bull. Lond. Math. Soc. 37 (2005), 25–36.CrossRefGoogle Scholar
[13]Rhin, G. and Wu, Q., ‘On the smallest value of the maximal modulus of an algebraic integer’, Math. Comp. 76 (2007), 1025–1038.CrossRefGoogle Scholar
[14]Schur, I., ‘Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten’, Math. Z. 1 (1918), 377–402.CrossRefGoogle Scholar
[15]Siegel, C. L., ‘The trace of totally positive and real algebraic integers’, Ann. of Math. (2) 46 (1945), 302–312.CrossRefGoogle Scholar
[16]Smyth, C. J., ‘On the measure of totally real algebraic integers’, J. Aust. Math. Soc. Ser. A 30 (1980), 137–149.CrossRefGoogle Scholar
[17]Smyth, C. J., ‘Totally positive algebraic integers of small trace’, Ann. Inst. Fourier (Grenoble) 34(3) (1984), 1–28.CrossRefGoogle Scholar
[18]Smyth, C. J., ‘The mean values of totally real algebraic integers’, Math. Comp. 42 (1984), 663–681.CrossRefGoogle Scholar
[19]Smyth, C. J., ‘Salem numbers of negative trace’, Math. Comp. 69 (2000), 827–838.CrossRefGoogle Scholar
[20]Wu, Q., ‘On the linear independence measure of logarithms of rational numbers’, Math. Comp. 72 (2003), 901–911.CrossRefGoogle Scholar