×

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)
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] Bundschuh, P. and Wylegala, F.-J. : Über algebraische Unabhängigkeit bei gewissen nichtfortsetzbaren Potenzreihen . Arch. Math. 34 (1980) 32-36. · Zbl 0414.10033
[2] Cijsouw, P.L. and Tijdeman, R. : On the transcendence of certain power series of algebraic numbers . Acta Arith. 23 (1973) 301-305. · Zbl 0255.10038
[3] Evertse, J.-H. : On sums of S-units and linear recurrences . Compositio Math. 53 (1984) 225-244. · Zbl 0547.10008
[4] Nishioka, K. : Algebraic independence of certain power series of algebraic numbers . J. Number Theory, to appear. · Zbl 0589.10035
[5] Nishioka, K. : Algebraic independence of three Liouville numbers . Arch. Math., to appear. · Zbl 0596.10036
[6] Nishioka, K. : Proof of Masser’s conjecture on the algebraic independence of values of Liouville series . Proc. Japan Acad. Ser. A 62 (1986) 219-222. · Zbl 0618.10032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.