Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The trace of totally positive algebraic integers

Author(s): Julián Aguirre; Mikel Bilbao; Juan Carlos Peral.
Journal: Math. Comp. 75 (2006), 385-393.
MSC (2000): Primary 11R06, 11-04
Posted: September 12, 2005
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: For all totally positive algebraic numbers $\alpha$ except a finite number of explicit exceptions, the following inequality holds:

\begin{displaymath}\frac{1}{d}\,(\alpha_1+\dots+\alpha_d)>\max(1.780022,1.66+\alpha_1), \end{displaymath}

where $d$ is the degree of $\alpha$ and $0<\alpha_1<\dots<\alpha_d$ its conjugates. This improves previous results of Smyth, Flammang and Rhin.

References:

[A]
Aparicio, B. On the Asymptotic Structure of the Polynomials of Minimal Diophantine Deviation from Zero, Journal of Approximation Theory 55 (1988), 270-278. MR 0968933 (90b:41010)

[BE]
Borwein, P. and Erdelyi, T. The integer Chebyshev problem, Math. Comp. 214 (1996), 661-681. MR 1333305 (96g:11077)

[FRS]
Flammang, V., Rhin, G., and Smyth, C.J. The integer transfinite diameter of intervals and totally real algebraic integers, J. Theor. Nombres-Bordeaux 9 (1997), 137-168. MR 1469665 (98g:11119)

[HS]
Habsieger, L. and Salvy, B. On integer Chebyshev polynomials, Math. Comp. 218 (1997), 763-770.MR 1401941 (97f:11053)

[R]
Ralston, A First Course in Numerical Analysis, Mc Graw-Hill, New York, 1965. MR 0191070 (32:8479)

[Si]
Siegel, C.L. The trace of totally positive and real algebraic integers, Ann. Math. 46 (1945), 302-312. MR 0012092 (6:257a)

[Sm1]
Smyth, C.J. Totally positive algebraic integers of small trace, Ann. Inst. Fourier Grenoble 34 (1984), 1-28. MR 0762691 (86f:11091)

[Sm2]
Smyth, C.J. The mean values of totally real algebraic integers, Math. of Comp. 42 (1984), 663-681. MR 0736460 (86e:11115)

[Sm3]
Smyth, C.J. An inequality for polynomials, CRM Proceedings and Lecture Notes 19 (1999), 315-321. MR 1684612 (2000d:11145)

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 11R06, 11-04

Retrieve articles in all Journals with MSC (2000): 11R06, 11-04

Additional Information:

Julián Aguirre
Affiliation: Departamento de Matemáticas, Universidad del Pa{í}s Vasco, Aptdo. 644, 48080 Bilbao, Spain
Email: mtpagesj@lg.ehu.es

Mikel Bilbao
Affiliation: Departamento de Econom{í}a Aplicada I, Universidad del Pa{í}s Vasco, Avda. Lehendakari Aguirre 83, 48015 Bilbao, Spain
Email: elpbillm@bs.ehu.es

Juan Carlos Peral
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain
Email: mtppealj@lg.ehu.es

DOI: 10.1090/S0025-5718-05-01776-X
PII: S 0025-5718(05)01776-X
Received by editor(s): July 2, 2004
Received by editor(s) in revised form: October 27, 2004
Posted: September 12, 2005
Copyright of article: Copyright 2005, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google