On the spectrum of the Zhang-Zagier height

作     者：Doche, C 

作者机构：Univ Bordeaux 1 Lab Algorithm Arithmet F-33405 Talence France 

出 版 物：《MATHEMATICS OF COMPUTATION》 (计算数学)

年 卷 期：2001年第70卷第233期

页      面：419-430页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：Mahler measure conjecture of Bogomolov 

摘      要：From recent work of Zhang and of Zagier, we know that their height h(alpha) is bounded away from 1 for every algebraic number alpha different from 0, 1;1/2 +/- root -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, rye describe an algorithm able to find them, and we give an algebraic number with height 1.2875274... discovered in this way. This search up to degree 64 suggests that the spectrum of h(alpha) may have a limit point less than 1.292. We prove this fact in the fourth part.