Proof of Masser’s conjecture on the algebraic independence of values of Liouville series. (English) Zbl 0618.10032

Nishioka, Kumiko

Proc. Japan Acad., Ser. A 62, 219-222 (1986).

The author proves the following conjecture of Masser on the algebraic independence of Liouville series: Let \(f(z)=\sum^{\infty}_{k=1}z^{k!}\) and let \(f^{(\ell)}(z)\) \((\ell =0,1,2,...)\) denote the \(\ell\)-th derivative of f. Suppose that \(\alpha_ 1,...,\alpha_ n\) are algebraic numbers such that \(0<| \alpha_ i| <1\) for \(i=1,...,n\) and no \(\alpha_ i/\alpha_ j\) \((1\leq i<j\leq n)\) is a root of unity. Then the numbers \(f^{(\ell)}(\alpha_ i)\) (1\(\leq i\leq n\), \(\ell \geq 0)\) are algebraically independent.
The proof uses a result of the reviewer on linear equations in S-units [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008)] which is a consequence of Schlickewei’s p-adic subspace theorem.

Reviewer: J.-H.Evertse

MSC:

11J85	Algebraic independence; Gel’fond’s method
30B10	Power series (including lacunary series) in one complex variable

Keywords:

algebraic independence; Liouville series

Citations:

Zbl 0547.10008

References:

[1]	J.-H. Evertse: On sums of S-units and linear recurrences. Comp. Math., 53, 225-244 (1984). · Zbl 0547.10008
[2]	K. Nishioka: Algebraic independence of certain power series of algebraic numbers (to appear in J. Number Theory). · Zbl 0589.10035
[3]	K. Nishioka: Algebraic independence of three Liouville numbers (to appear in Arch. Math.). · Zbl 0596.10036