Unit Equations in Diophantine Number Theory (Cambridge Studies in Advanced Mathematics, Series Number 146)

Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field. Part 1. Preliminaries. Basic Algebraic Number Theory -- Algebraic Function Fields -- Tools From Diophantine Approximation And Transcendence Theory -- Part 2. Unit Equations And Applications. Effective Results For Unit Equations In Two Unknowns Over Number Fields -- Algorithmic Resolution Of Unit Equations In Two Unknowns -- Unit Equations In Several Unknowns -- Analogues Over Function Fields -- Effective Results For Unit Equations Over Finitely Generated Domains -- Decomposable Form Equations -- Further Applications. Jan-hendrik Evertse, Universiteit Leiden, Kálmán Györy, Debreceni Egyetem, Hungary. Includes Bibliographical References (pages 337-357) And Index. cover title copy write Contents Preface Summary 1 Basic algebraic number theory 2 Algebraic function fields 3 Tools from Diophantine approximation and transcendence theory 4 Effective results for unit equations in two unknowns over number fields 5 Algorithmic resolution of unit equations in two unknowns 6 Unit equations in several unknowns 7 Analogues over function fields 8 Effective results for unit equations in two unknowns over finitely generated domains 9 Decomposable form equations 10 Further applications References Glossary of frequently used notation Index Unit equations play a central role in Diophantine number theory. This book provides a comprehensive and up-to-date treatment of unit equations and their various applications. It brings together the most important results and gives an overview of the basic techniques, making it accessible to young researchers.