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Sous-groupes minimaux des groupes de Lie commutatifs réels, et applications arithmétiques. (Minimal subgroups of commutative real Lie groups and arithmetic applications). (French) Zbl 0672.10024

Let R be a commutative real Lie group. We say that a subgroup \(\Gamma\) of R is a minimal subgroup of R if it is dense in R, finitely generated, and if no subgroup of \(\Gamma\) of rank strictly less than the rank of \(\Gamma\) is dense in R. In this paper, we show that the rank of such a subgroup is bounded by a constant which depends only on R. In the case where R is \(({\mathbb{R}}^{\times})^ p\times ({\mathbb{C}}^{\times})^ q,\) this bound is \(2p+3q.\) The main result is that if \(\Gamma\) is a minimal subgroup of \(({\mathbb{R}}^{\times})^ p\times ({\mathbb{C}}^{\times})^ q\) contained in \(({\bar {\mathbb{Q}}}^{\times}\cap {\mathbb{R}}^{\times})^ p\times ({\bar {\mathbb{Q}}}^{\times})^ q\) then its rank is \(p+q+1\) for \(q=0\), \(1\leq p\leq 4\) and for \(q=1\), \(p\leq 1\). This gives a partial answer to a question of J. J. Sansuc.
Reviewer: D.Roy

MSC:

11J72 Irrationality; linear independence over a field
11J81 Transcendence (general theory)
22E15 General properties and structure of real Lie groups
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