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Algebraic independence of transcendental numbers. Gel’fond’s method and its developments. (English) Zbl 0556.10023

Perspectives in mathematics, Anniv. Oberwolfach 1984, 551-571 (1984).
[For the entire collection see Zbl 0548.00010.]
The present survey paper gives an excellent account to the method of A. O. Gel’fond [Usp. Mat. Nauk 4, No. 5(33), 14–48 (1949; Zbl 0039.04401)] which until now is one of the very few more general methods for algebraic independence proofs. The author traces the most important developments of this method during the last 15 years, when its most significant generalizations, refinements, applications etc. were discovered. More precisely the text is subdivided as follows.
I. Small transcendence degree: How to produce fields generated by values of exponential or elliptic functions, for which the transcendence degree is at least two or three?
II. Large transcendence degree: Same question as in I, but with transcendence degree \(\geq n\), where \(n\) is large; the three known approaches due to Chudnovsky, Masser-Wüstholz, and Philippon are discussed.
III. Criteria of algebraic independence: Gel’fond’s original criterion for one variable, its generalization to two variables, the criteria of Chudnovsky-Reyssat and Philippon.
IV. Further conjectures (on exponential and elliptic functions as well as on algebraic groups).
{Reviewer’s remarks: In line 20 on p. 554 [\({\mathbb Q}(\beta):{\mathbb Q}]\) has to be 3, not 2. As D. W. Masser has indicated, the effective version of Schanuel’s conjecture on p. 556 contradicts results of A. Bijlsma [Compos. Math. 35, 99–111 (1977; Zbl 0355.10025)].}
Reviewer: P.Bundschuh

MSC:

11J85 Algebraic independence; Gel’fond’s method
11-02 Research exposition (monographs, survey articles) pertaining to number theory