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On the spectrum of the Zhang-Zagier height

Author: Christophe Doche
Journal: Math. Comp. 70 (2001), 419-430
MSC (2000): Primary 11R04, 11R06; Secondary 12D10
Published electronically: March 3, 2000
MathSciNet review: 1681120
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From recent work of Zhang and of Zagier, we know that their height $\mathfrak{H}(\alpha)$ is bounded away from 1 for every algebraic number $\alpha$ different from $0,1,1/2\pm\sqrt{-3}/2$. The study of the related spectrum is especially interesting, for it is linked to Lehmer's problem and to a conjecture of Bogomolov. After recalling some definitions, we show an improvement of the so-called Zhang-Zagier inequality. To achieve this, we need some algebraic numbers of small height. So, in the third section, we describe an algorithm able to find them, and we give an algebraic number with height $1.2875274\ldots$ discovered in this way. This search up to degree 64 suggests that the spectrum of $\mathfrak{H}(\alpha)$ may have a limit point less than 1.292. We prove this fact in the fourth part.

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  • [B78] D. W. Boyd, Variations on a theme of Kronecker, Canad. Math. Bull. 21 (1978), 129-133. MR 58:5580
  • [B80] -, Reciprocal polynomials having small measure, Math. Comp. 35 (1980), 1361-1377. MR 82a:30005
  • [B89] -, Reciprocal polynomials having small measure II, Math. Comp. 53 (1989), 355-357. MR 89m:30013
  • [DP98] S. David and P. Philippon, Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes. (French) Number Theory, 333-364, Contemp. Math., 210, Amer. Math. Soc., Providence, RI, 1998. MR 98j:11044
  • [DHJ95] J. Dégot, J.-C. Hohl and O. Jenvrin, Calcul numérique de la mesure de Mahler d'un polynôme par itérations de Graeffe, C.R. Acad. Sci. Paris 320 I (1995), 269-272. MR 96e:11038
  • [D98] G. P. Dresden, Orbits of algebraic numbers with low heights, Math. Comp. 67 (1998), 815-820. MR 98h:11128
  • [F94] V. Flammang, Mesures de polynômes. Applications au diamètre transfini entier, Thèse de l'Université de Metz, 1994. MR 97d:68003
  • [GKP94] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete mathematics, second edition, Addison-Wesley, 1994.
  • [M95] M. J. Mossinghoff, Algorithms for the determination of polynomials with small Mahler measure, Ph.D. Thesis, The University of Texas at Austin, 1995.
  • [RS97] G. Rhin and C. J. Smyth, On the Mahler measure of the composition of two polynomials, Acta. Arith. 79 (1997), 239-247. MR 98b:11109
  • [Si95] J. H. Silverman, Exceptional units and numbers of small Mahler measure, Experiment. Math. 4 (1995) N $^{\mathrm o}1$, 69-83. MR 96j:11150
  • [Sm71] C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175. MR 44:6641
  • [Sm81] -, On the measure of totally real algebraic integers II, Math. Comp. 37 (1981), 205-208. MR 82j:12002b
  • [U98] E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. 147 (1998), 167-179. MR 99e:14031
  • [Za93] D. Zagier, Algebraic numbers close both to 0 and 1, Math. Comp. 61 (1993), 485-491. MR 94c:11104
  • [Zh92] S. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. 136 (1992), 569-587. MR 93j:14024

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Additional Information

Christophe Doche
Affiliation: Laboratoire d’Algorithmique Arithmétique, Université Bordeaux I, 351 cours de la Libération, F-33405 Talence Cedex France

Keywords: Mahler measure, conjecture of Bogomolov
Received by editor(s): October 23, 1998
Received by editor(s) in revised form: February 2, 1999
Published electronically: March 3, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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