Totally positive algebraic integers with small trace
HTML articles powered by AMS MathViewer
Abstract:
The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for $S_k$, and reduce sharply the computing time. We are then able to push the computation to degree $15$ and prove that there is no such totally positive algebraic integer with absolute trace $1.8$. As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to $1.793145\cdots$.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 integer Chebyshev constant of Farey intervals, Publ. Mat. Proceedings of the Primeras Jornadas de Teoría de Números (2007), 11–27. MR 2499685, DOI 10.5565/PUBLMAT_{P}JTN05_{0}1
- 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
- Edward J. Anderson and Peter Nash, Linear programming in infinite-dimensional spaces, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Ltd., Chichester, 1987. Theory and applications; A Wiley-Interscience Publication. MR 893179
- Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495, DOI 10.1007/978-0-387-21652-2
- 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
- David W. Boyd, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985), no. 171, 243–249, S17–S20. MR 790657, DOI 10.1090/S0025-5718-1985-0790657-8
- Youyan Chen, Chenggang Peng, and Qiang Wu, Finding all Salem numbers of trace $-1$ and degree up to 20, Taiwanese J. Math. 22 (2018), no. 1, 23–37. MR 3749352, DOI 10.11650/tjm/8208
- Xiaoqian Dong and Qiang Wu, The absolute trace of totally positive reciprocal algebraic integers, J. Number Theory 170 (2017), 66–74. MR 3541699, DOI 10.1016/j.jnt.2016.06.016
- 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, Une nouvelle minoration pour la trace absolue des entiers algébriques totalement positifs, hal-01346165 (2016).
- 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, Georges Rhin, and Jean-Marc Sac-Épée, Integer transfinite diameter and polynomials with small Mahler measure, Math. Comp. 75 (2006), no. 255, 1527–1540. MR 2219043, DOI 10.1090/S0025-5718-06-01791-1
- A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
- Yanhua Liang and Qiang Wu, The trace problem for totally positive algebraic integers, J. Aust. Math. Soc. 90 (2011), no. 3, 341–354. MR 2833305, DOI 10.1017/S1446788711001030
- James McKee, Computing totally positive algebraic integers of small trace, Math. Comp. 80 (2011), no. 274, 1041–1052. MR 2772109, DOI 10.1090/S0025-5718-2010-02424-X
- 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
- Michael J. Mossinghoff, Georges Rhin, and Qiang Wu, Minimal Mahler measures, Experiment. Math. 17 (2008), no. 4, 451–458. MR 2484429, DOI 10.1080/10586458.2008.10128872
- S. El Otmani, A. Maul, G. Rhin, and J.-M. Sac-Épée, Finding degree-16 monic irreducible integer polynomials of minimal trace by optimization methods, Exp. Math. 23 (2014), no. 1, 1–5. MR 3177452, DOI 10.1080/10586458.2013.849213
- S. El Otmani, G. Rhin, and J.-M. Sac-Épée, A Salem number with degree 34 and trace $-3$, J. Number Theory 150 (2015), 21–25. MR 3304603, DOI 10.1016/j.jnt.2014.11.013
- Georges Rhin and Qiang Wu, On the smallest value of the maximal modulus of an algebraic integer, Math. Comp. 76 (2007), no. 258, 1025–1038. MR 2291848, DOI 10.1090/S0025-5718-06-01958-2
- 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.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, Salem numbers of negative trace, Math. Comp. 69 (2000), no. 230, 827–838. MR 1648407, DOI 10.1090/S0025-5718-99-01099-6
- 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
- Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010), no. 272, 2387–2394. MR 2684371, DOI 10.1090/S0025-5718-10-02345-8
Additional Information
- Cong Wang
- Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road, Beibei, 400715 Chongqing, People’s Republic of China
- Email: wangcong.swu@foxmail.com
- Jie Wu
- Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road, Beibei, 400715 Chongqing, People’s Republic of China
- Address at time of publication: CNRS, UMR 8050, Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France
- ORCID: 0000-0002-6893-7938
- Email: jie.wu@math.cnrs.fr
- Qiang Wu
- Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road, Beibei, 400715 Chongqing, People’s Republic of China
- Email: qiangwu@swu.edu.cn
- Received by editor(s): October 26, 2020
- Received by editor(s) in revised form: December 30, 2020
- Published electronically: May 20, 2021
- Additional Notes: This work was supported in part by the National Natural Science Foundation of China (Grant no. 12071375) and the NSF of Chongqing (Grant No. cstc2019jcyj-msxm1651).
The third author is the corresponding author. - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 2317-2332
- MSC (2020): Primary 11C08, 11R06, 11Y40
- DOI: https://doi.org/10.1090/mcom/3636
- MathSciNet review: 4280303