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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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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