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Nicolas Ratazzi
Problème de Lehmer sur ${\mathbb{G}_m}$ et méthode des pentes
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 231-248, doi: 10.5802/jtnb.584
Article PDF | Reviews MR 2332064 | Zbl pre05186985

Résumé - Abstract

Let $h$ be the usual absolute logarithmic Weil height on $\overline{\mathbb{Q}}^{\times }$. Using the slopes inequality of J.-B. Bost, we give in this article a proof of the following result of Dobrowolski [4] : there exists a constant $c>0$ such that

$$\forall x\in \mathbb{G}_m(\overline{\mathbb{Q}})\backslash \mu _{\infty } \ \ h(x)\ge \frac{c}{D}\left(\frac{\log \log 3D}{\log 2D}\right)^{3},$$

where $D=[\mathbb{Q}(x) : \mathbb{Q}]$ and where $\mu _{\infty }$ denote the group of roots of unity.

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