Proof of Masser's conjecture on the algebraic independence of values of Liouville series
type: Item
instance of: scholarly article
title: Proof of Masser's conjecture on the algebraic independence of values of Liouville series
zbMATH Open document ID: 0618.10032
author: Kumiko Nishioka
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11J85
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 4002191
zbMATH Keywords: algebraic independence
zbMATH Keywords: Liouville series
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
full work available at URL: pjaa.62.219
OpenAlex ID: W2030563625
Wikidata QID: Q123005498
cites work: Q3338318
cites work: Algebraic independence of certain power series of algebraic numbers
cites work: Algebraic independence of three Liouville series
DOI: 10.3792/PJAA.62.219
description: scientific article; zbMATH DE number 4002191