Skip to main content Accessibility help
×
Home

THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

  • YANHUA LIANG (a1) and QIANG WU (a2)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS
      Available formats
      ×

Copyright

Corresponding author

For correspondence; e-mail: qiangwu@swu.edu.cn

Footnotes

Hide All

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

Footnotes

References

Hide All
[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. 1119, With an appendix by Jean-Pierre Serre.
[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).
[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.
[4]Borwein, P., Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics, 10 (Springer, New York, 2002).
[5]Flammang, V., ‘Trace of totally positive algebraic integers and integer transfinite diameter’, Math. Comp. 78 (2009), 11191125.
[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. 165168.
[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.
[8]Flammang, V., Rhin, G. and Sac-Épée, J. M., ‘Integer transfinite diameter and polynomials of small Mahler measure’, Math. Comp. 75 (2006), 15271540.
[9]Habsieger, L. and Salvy, B., ‘On integer Chebyshev polynomials’, Math. Comp. 66 (1997), 763770.
[10]McKee, J., ‘Computing totally positive algebraic integers of small trace’, Math. Comp. 80 (2011), 10411052.
[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. 327337.
[12]McKee, J. and Smyth, C. J., ‘There are Salem numbers of every trace’, Bull. Lond. Math. Soc. 37 (2005), 2536.
[13]Rhin, G. and Wu, Q., ‘On the smallest value of the maximal modulus of an algebraic integer’, Math. Comp. 76 (2007), 10251038.
[14]Schur, I., ‘Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten’, Math. Z. 1 (1918), 377402.
[15]Siegel, C. L., ‘The trace of totally positive and real algebraic integers’, Ann. of Math. (2) 46 (1945), 302312.
[16]Smyth, C. J., ‘On the measure of totally real algebraic integers’, J. Aust. Math. Soc. Ser. A 30 (1980), 137149.
[17]Smyth, C. J., ‘Totally positive algebraic integers of small trace’, Ann. Inst. Fourier (Grenoble) 34(3) (1984), 128.
[18]Smyth, C. J., ‘The mean values of totally real algebraic integers’, Math. Comp. 42 (1984), 663681.
[19]Smyth, C. J., ‘Salem numbers of negative trace’, Math. Comp. 69 (2000), 827838.
[20]Wu, Q., ‘On the linear independence measure of logarithms of rational numbers’, Math. Comp. 72 (2003), 901911.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

  • YANHUA LIANG (a1) and QIANG WU (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed