Proof of Masser's conjecture on the algebraic independence of values of Liouville series
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.
scientific article; zbMATH DE number 4002191