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Zbl 1011.11045
Amoroso, Francesco; David, Sinnou
The higher-dimensional Lehmer problem. (Le problème de Lehmer en dimension supérieure.) (French)
[J] J. Reine Angew. Math. 513, 145-179 (1999). ISSN 0075-4102; ISSN 1435-5345/e

Summary: We study a higher-dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if $\alpha_1,\dots, \alpha_n$ are multiplicatively independent algebraic numbers, we provide a lower bound for the product of the heights of the $\alpha_i$'s in terms of the degree $D$ of the number field generated by the $\alpha_i$'s. This enables us to study the successive minima for the height function in a given number field. Our bound is a generalization of an earlier result of {\it E. Dobrowolski} [Acta Arith. 34, 391--401 (1979; Zbl 0416.12001)] and is best possible up to a power of $\log(D)$. This, in particular, shows that the Lehmer problem is true for number fields having a `small' Galois group. \par The main result bases on two theorems, first an analogue of Dobrowolski's key lemma and second a version of Philippon's zero estimates with multiplication.

MSC 2000:: *11G50 Heights
11J95 Results of diophantine approximation involving abelian varieties
14G40 Arithmetic varieties and schemes

Keywords: Siegel lemma; extrapolation; rank estimate; higher-dimensional Lehmer problem; power of the multiplicative group; lower bound; heights; successive minima for the height function

Citations: Zbl 0416.12001

Cited in: Zbl 1194.11070 Zbl 1150.11021 Zbl 1041.11048 Zbl 0938.11031

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