## 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

### Keywords:

algebraic independence; Liouville series

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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