## Computing totally positive algebraic integers of small trace

HTML articles powered by AMS MathViewer

- by James McKee;
- Math. Comp.
**80**(2011), 1041-1052 - DOI: https://doi.org/10.1090/S0025-5718-2010-02424-X
- Published electronically: October 22, 2010
- PDF | Request permission

Table supplement: Some minimal trace totally positive algebraic integers (PDF)

## Abstract:

We construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials. Many new examples of such polynomials of minimal absolute trace (for given degree) are found. The computations are pushed to degrees that previously were unattainable, and one consequence is that the new examples form the majority of all those known. As an application, we produce a new bound for the Schur-Siegel-Smyth trace problem.## References

- Julián Aguirre, Mikel Bilbao, and Juan Carlos Peral,
*The trace of totally positive algebraic integers*, Math. Comp.**75**(2006), no. 253, 385–393. MR**2176405**, DOI 10.1090/S0025-5718-05-01776-X - Julián Aguirre and Juan Carlos Peral,
*The trace problem for totally positive algebraic integers*, Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 1–19. With an appendix by Jean-Pierre Serre. MR**2428512**, DOI 10.1017/CBO9780511721274.003 - Henri Cohen,
*A course in computational algebraic number theory*, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR**1228206**, DOI 10.1007/978-3-662-02945-9 - V. Flammang,
*Trace of totally positive algebraic integers and integer transfinite diameter*, Math. Comp.**78**(2009), no. 266, 1119–1125. MR**2476574**, DOI 10.1090/S0025-5718-08-02120-0 - V. Flammang, M. Grandcolas, and G. Rhin,
*Small Salem numbers*, Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 165–168. MR**1689505** - Valérie Flammang and Georges Rhin,
*Algebraic integers whose conjugates all lie in an ellipse*, Math. Comp.**74**(2005), no. 252, 2007–2015. MR**2164108**, DOI 10.1090/S0025-5718-05-01735-7 - James McKee and Chris Smyth,
*Salem numbers of trace $-2$ and traces of totally positive algebraic integers*, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 327–337. MR**2137365**, DOI 10.1007/978-3-540-24847-7_{2}5 - James McKee and Chris Smyth,
*There are Salem numbers of every trace*, Bull. London Math. Soc.**37**(2005), no. 1, 25–36. MR**2105815**, DOI 10.1112/S0024609304003790 - The PARI Group, PARI version 2.3.4, Bordeaux 2006, available from http://pari.math.u-bordeaux.fr/.
- Raphael M. Robinson,
*Algebraic equations with span less than $4$*, Math. Comp.**18**(1964), 547–559. MR**169374**, DOI 10.1090/S0025-5718-1964-0169374-X - I. Schur,
*Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten*, Math. Z.**1**(1918), no. 4, 377–402 (German). MR**1544303**, DOI 10.1007/BF01465096 - 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 - —, An inequality for polynomials,
*Number theory*(Ottawa, ON, 1996), CRM Proc. Lecture Notes**19**, 315–321, Amer. Math. Soc., Providence, RI, 1999. - Qiang Wu,
*On the linear independence measure of logarithms of rational numbers*, Math. Comp.**72**(2003), no. 242, 901–911. MR**1954974**, DOI 10.1090/S0025-5718-02-01442-4

## Bibliographic Information

**James McKee**- Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, England, United Kingdom
- Email: james.mckee@rhul.ac.uk
- Received by editor(s): December 23, 2009
- Received by editor(s) in revised form: February 22, 2010
- Published electronically: October 22, 2010
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**80**(2011), 1041-1052 - MSC (2010): Primary 11R04, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-2010-02424-X
- MathSciNet review: 2772109