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Conditions for algebraic independence of certain power series of algebraic numbers. (English) Zbl 0615.10042
The main result proved by the author may be resumed as follows. Let f be a power series satisfying certain growth condition on the indices of the non null coefficients. Let $$\alpha_ 1,...,\alpha_ n$$ be algebraic numbers in the convergence disk of f. Then the numbers 1, $$f(\alpha_ 1),...,f(\alpha_ n)$$ are linearly dependent over the algebraic closure of $${\mathbb{Q}}$$ if and only if the $$f^{(\ell)}(\alpha_ i)$$ are linearly dependent over $${\mathbb{Q}}$$, $$1\leq i\leq n$$, $$\ell \in {\mathbb{N}}$$. The method requires partial derivatives of polynomials in a great number of variables.
As an example the author proves the following conjecture due to Masser: Let $$f(z)=\sum z^{k!}$$ and let $$\alpha_ 1,...,\alpha_ n$$ be algebraic numbers $$(| \alpha_ i| <1)$$. Then for each $$\ell$$, the $$f^{(\ell)}(\alpha_ i)$$ are algebraically independent if and only if the $$\alpha_ i/\alpha_ j$$ are not roots of unity (for $$i\neq j).$$
Also the author gives a p-adic version of his theorem that only involves the case $$\ell =0$$. One can hope the result holds in the p-adic field for all $$\ell$$, and that remains an interesting question.
Reviewer: A.Escassut

MSC:
 11J81 Transcendence (general theory)
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References:
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