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The higher-dimensional Lehmer problem. (Le problème de Lehmer en dimension supérieure.) (French) Zbl 1011.11045

Summary: We study a higher-dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if ${\alpha }_{1},\cdots ,{\alpha }_{n}$ 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 E. Dobrowolski [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)] and is best possible up to a power of $log\left(D\right)$. This, in particular, shows that the Lehmer problem is true for number fields having a ‘small’ Galois group.

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:
 11G50 Heights 11J95 Results of diophantine approximation involving abelian varieties 14G40 Arithmetic varieties and schemes; Arakelov theory; heights