## The trace of totally positive algebraic integers

HTML articles powered by AMS MathViewer

- by Julián Aguirre, Mikel Bilbao and Juan Carlos Peral;
- Math. Comp.
**75**(2006), 385-393 - DOI: https://doi.org/10.1090/S0025-5718-05-01776-X
- Published electronically: September 12, 2005
- PDF | Request permission

## Abstract:

For all totally positive algebraic numbers $\alpha$ except a finite number of explicit exceptions, the following inequality holds: \[ \frac {1}{d} (\alpha _1+\dots +\alpha _d)>\max (1.780022,1.66+\alpha _1), \] 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

- Emiliano Aparicio Bernardo,
*On the asymptotic structure of the polynomials of minimal Diophantic deviation from zero*, J. Approx. Theory**55**(1988), no. 3, 270–278. MR**968933**, DOI 10.1016/0021-9045(88)90093-7 - Peter Borwein and Tamás Erdélyi,
*The integer Chebyshev problem*, Math. Comp.**65**(1996), no. 214, 661–681. MR**1333305**, DOI 10.1090/S0025-5718-96-00702-8 - V. Flammang, G. Rhin, and C. J. Smyth,
*The integer transfinite diameter of intervals and totally real algebraic integers*, J. Théor. Nombres Bordeaux**9**(1997), no. 1, 137–168 (English, with English and French summaries). MR**1469665**, DOI 10.5802/jtnb.193 - Laurent Habsieger and Bruno Salvy,
*On integer Chebyshev polynomials*, Math. Comp.**66**(1997), no. 218, 763–770. MR**1401941**, DOI 10.1090/S0025-5718-97-00829-6 - Anthony Ralston,
*A first course in numerical analysis*, McGraw-Hill Book Co., New York-Toronto-London, 1965. MR**191070** - Carl Ludwig Siegel,
*The trace of totally positive and real algebraic integers*, Ann. of Math. (2)**46**(1945), 302–312. MR**12092**, DOI 10.2307/1969025 - Christopher Smyth,
*Totally positive algebraic integers of small trace*, Ann. Inst. Fourier (Grenoble)**34**(1984), no. 3, 1–28 (English, with French summary). MR**762691**, DOI 10.5802/aif.985 - C. J. Smyth,
*The mean values of totally real algebraic integers*, Math. Comp.**42**(1984), no. 166, 663–681. MR**736460**, DOI 10.1090/S0025-5718-1984-0736460-5 - C. J. Smyth,
*An inequality for polynomials*, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 315–321. MR**1684612**, DOI 10.1090/crmp/019/28

## Bibliographic Information

**Julián Aguirre**- Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain
- Email: mtpagesj@lg.ehu.es
**Mikel Bilbao**- Affiliation: Departamento de Economía Aplicada I, Universidad del País 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
- MR Author ID: 137825
- Email: mtppealj@lg.ehu.es
- Received by editor(s): July 2, 2004
- Received by editor(s) in revised form: October 27, 2004
- Published electronically: September 12, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp.
**75**(2006), 385-393 - MSC (2000): Primary 11R06, 11-04
- DOI: https://doi.org/10.1090/S0025-5718-05-01776-X
- MathSciNet review: 2176405