Effective bounds for solutions of \(S\)-unit equations and of Thue-Mahler equations
type: Item
instance of: scholarly article
title: Effective bounds for solutions of \(S\)-unit equations and of Thue-Mahler equations
zbMATH Open document ID: 0926.11015
author: Yann Bugeaud
published in: Journal of Number Theory
publication date: 1999-12-02T00:00:00Z
review text: \textit{E. Bombieri} and \textit{P. B. Cohen} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 205-225 (1997; Zbl 0912.11028)] have developed a method in effective diophantine approximation, based on a reworked version of the Thue principle, the Dyson lemma and on the geometry of numbers. The present author sharpens the results of the above paper by reworking it and using, instead of the Thue principle, lower bounds of linear forms in two or three logarithms. As a result, he gives effective upper bounds for the size of solutions of \(S\)-unit equations and of Thue-Mahler equations. He also derives a new effective irrationality measure for algebraic numbers. Finally, he compares his estimates with analogous recent estimates of himself and \textit{K. Győry} [Acta Arith. 74, 67-80 (1996; Zbl 0861.11023) and 273-292 (1996; Zbl 0861.11024)].
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 1236181
zbMATH Keywords: linear form in logarithms
zbMATH Keywords: upper bounds
zbMATH Keywords: solutions of \(S\)-unit equations
zbMATH Keywords: Thue-Mahler equations
zbMATH Keywords: effective irrationality measure for algebraic numbers
reviewed by: Nikos Tzanakis
MaRDI profile type: Publication
full work available at URL: jnth.1998.2245
OpenAlex ID: W1968620667
cites work: RATIONAL APPROXIMATIONS TO 23 AND OTHER ALGEBRAIC NUMBERS
cites work: Linear forms in the logarithms of algebraic numbers
cites work: Q3803198
cites work: Logarithmic forms and group varieties.
cites work: On the Thue-Siegel-Dyson theorem
cites work: Q4033105
cites work: Q4371383
cites work: Q3663376
cites work: Q4339612
cites work: On Siegel's lemma
cites work: Bounds for the solutions of unit equations
cites work: Bounds for the solutions of Thue-Mahler equations and norm form equations
cites work: Effective lower bound for the \(p\)-adic distance between powers of algebraic numbers
cites work: On the method of Thue-Siegel
cites work: Q5646967
cites work: Rational approximations to algebraic numbers
cites work: Diophantine approximation
cites work: An effective lower bound for the height of algebraic numbers
cites work: Q4296321
DOI: 10.1006/JNTH.1998.2245
description: scientific article; zbMATH DE number 1236181