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Sur les mesures d’indépendance algébrique. (On measures of algebraic independence). (French) Zbl 0567.10034

Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1983-84, Prog. Math. 59, 219-233 (1985).
[For the entire collection see Zbl 0561.00004.]
The author’s aim is to give a new algebraic independence measure for certain finite families of numbers improving Nesterenko’s results. Let K be a number field. Let v be a place in K and let \({\mathbb{C}}_ v\) be the completion of an algebraic closure of \(K_ v\). The author first gives a theorem deduced from his last article [Critères d’indépendance algébrique (Preprint École Polytechnique 1984) (to appear in Publ. Math., Inst. Hautes Étud. Sci.)]: Provided \(K[x_ 1,...,x_ n]\) has a sequence of ideals \(I_ n\) satisfying conditions on the set of the common zeros and the size of polynomials generating \(I_ n\), he can give a lower bound for the transcendence degree of \({\mathbb{Q}}(\theta_ 1,...,\theta_ n)\) when \(\theta_ i\in {\mathbb{C}}_ v\) (1\(\leq i\leq n).\)
The second theorem, too much technical to be summarized, gives, as a corollary, an algebraic independence measure for the family \((\alpha,\alpha^{\beta},...,\alpha^{\beta^{d}})\) in \({\mathbb{C}}^ d\) when \(\alpha\) is algebraic (different of 0 and 1) and \(\beta\) is algebraic of degree \(d\geq 2\).
Reviewer: A.Escassut

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J81 Transcendence (general theory)