Linear forms in logarithms and applications
type: Item
instance of: scholarly article
title: Linear forms in logarithms and applications
zbMATH Open document ID: 1394.11001
DOI: 10.4171/183
author: Yann Bugeaud
published in: IRMA Lectures in Mathematics and Theoretical Physics
publication date: 2018-02-21T00:00:00Z
review text: The book is a collection of the present results on linear forms in the logarithms of algebraic numbers, together with its applications, useful both for researchers and students. The classical chapter of the theory of Diophantine number theory was closed in 1966 and a new chapter was opened as professor Alan Baker published his revolutionary new ideas in mathematics about linear forms in the logarithms of algebraic numbers, establishing his effective methods. Short afterwards in 1971, he was awarded the Fields Medal for his works, that influenced wide parts of number theory. The book of Yann Bugeaud is a summary of Baker's theory and its applications. The book is carefully written to include all details and to present them in a way, that can be very well used both by researchers and by interested university students. The author presents results on linear form in complex logarithms, \(p\)-adic logarithms and elliptic logarithms, involves also theorems on simultaneous linear forms in logarithms. Special estimates on linear forms in two logarithms are discussed in separate chapters. The book includes applications of Baker's theory to some well-known types of Diophantine equations, such like Thue equations, superelliptic and hyperelliptic equations, Ramanujan-Nagell equation, Catalan equation, S-unit equations. Among other topics, the greatest prime factor of \(2^n-1\) is discussed, as well as the relation of Baker's theory to \(abc\)-conjecture. The book is completed with an appendix, containing the basic knowledge on Diophantine approximation, algebraic number fields, heights, prime number theory, and even tools from complex and \(p\)-adic calculus. An extensive bibliography in given.
reviewed by: István Gaál
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11Dxx
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 6841731
zbMATH Keywords: linear forms in the logarithms of algebraic numbers
zbMATH Keywords: transcendence theory
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: Diophantine equations
MaRDI profile type: Publication
OpenAlex ID: W2785641372
description: scientific article; zbMATH DE number 6841731