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

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

Citations:

Zbl 0547.10008
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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
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