On the spectrum of the Zhang-Zagier height

Doche, Christophe

doi:10.1090/S0025-5718-00-01183-2

On the spectrum of the Zhang-Zagier height
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by Christophe Doche;

Math. Comp. 70 (2001), 419-430

DOI: https://doi.org/10.1090/S0025-5718-00-01183-2

Published electronically: March 3, 2000

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Abstract:

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.

References

David W. Boyd, Variations on a theme of Kronecker, Canad. Math. Bull. 21 (1978), no. 2, 129–133. MR 485771, DOI 10.4153/CMB-1978-023-x
David W. Boyd, Reciprocal polynomials having small measure, Math. Comp. 35 (1980), no. 152, 1361–1377. MR 583514, DOI 10.1090/S0025-5718-1980-0583514-9
David W. Boyd, Reciprocal polynomials having small measure. II, Math. Comp. 53 (1989), no. 187, 355–357, S1–S5. MR 968149, DOI 10.1090/S0025-5718-1989-0968149-6
Sinnou David and Patrice Philippon, Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes, Number theory (Tiruchirapalli, 1996) Contemp. Math., vol. 210, Amer. Math. Soc., Providence, RI, 1998, pp. 333–364 (French, with English and French summaries). MR 1478502, DOI 10.1090/conm/210/02795
Jérôme Dégot, Jean-Christophe Hohl, and Odile Jenvrin, Calcul numérique de la mesure de Mahler d’un polynôme par itérations de Graeffe, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 269–272 (French, with English and French summaries). MR 1320369
Gregory P. Dresden, Orbits of algebraic numbers with low heights, Math. Comp. 67 (1998), no. 222, 815–820. MR 1468942, DOI 10.1090/S0025-5718-98-00963-6
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete mathematics, second edition, Addison-Wesley, 1994.
M. J. Mossinghoff, Algorithms for the determination of polynomials with small Mahler measure, Ph.D. Thesis, The University of Texas at Austin, 1995.
G. Rhin and C. J. Smyth, On the Mahler measure of the composition of two polynomials, Acta Arith. 79 (1997), no. 3, 239–247. MR 1438826, DOI 10.4064/aa-79-3-239-247
Joseph H. Silverman, Exceptional units and numbers of small Mahler measure, Experiment. Math. 4 (1995), no. 1, 69–83. MR 1359419, DOI 10.1080/10586458.1995.10504309
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 289451, DOI 10.1112/blms/3.2.169
C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 137–149. MR 607924, DOI 10.1017/S1446788700016426
Emmanuel Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167–179 (French). MR 1609514, DOI 10.2307/120987
D. Zagier, Algebraic numbers close to both $0$ and $1$, Math. Comp. 61 (1993), no. 203, 485–491. MR 1197513, DOI 10.1090/S0025-5718-1993-1197513-9
Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569–587. MR 1189866, DOI 10.2307/2946601