The trace of totally positive algebraic integers
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- by Julián Aguirre, Mikel Bilbao and Juan Carlos Peral PDF
- Math. Comp. 75 (2006), 385-393 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 0191070
- 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.975
- 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
Additional 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