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.
- Presents the state of the art following enormous progress over the past forty years
- Compiles results previously scattered across the literature
- An important resource for researchers working in Diophantine number theory
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Reviews & endorsements
'The book is well written and is certain to be of use to experts and graduate students alike. Every chapter is prefaced with a nice introduction and summary, putting the material in perspective and surveying the proof techniques. The book also features an extensive bibliography and an easy-to-use glossary and index.' Lenny Fukshansky, MathSciNet
'Understanding the book requires only basic knowledge in algebra (groups, commutative rings, fields, Galois theory and elementary algebraic number theory). In particular the concepts of height, places and valuations play an important role. … In addition to providing a survey, the authors improve both formulations and proofs of various important existing results in the literature, making their book a valuable asset for researchers in this area. … I thank them for their fast and clear answers.' Pieter Moree, Nieuw Archief voor Wiskunde
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Product details
- Date Published: January 2016
- format: Hardback
- isbn: 9781107097605
- length: 378 pages
- dimensions: 235 x 158 x 25 mm
- weight: 0.66kg
- availability: Available
Table of Contents
Preface
Summary
Glossary of frequently used notation
Part I. Preliminaries:
1. Basic algebraic number theory
2. Algebraic function fields
3. Tools from Diophantine approximation and transcendence theory
Part II. Unit equations and applications:
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 over finitely generated domains
9. Decomposable form equations
10. Further applications
References
Index.
Look Inside
Authors
Jan-Hendrik Evertse, Universiteit Leiden
Jan-Hendrik Evertse is Assistant Professor in the Mathematical Institute at Leiden University. His research concentrates on Diophantine approximation and applications to Diophantine problems. In this area he has obtained some influential results, in particular on estimates for the numbers of solutions of Diophantine equations and inequalities. He has written more than 75 research papers and co-authored one book with Bas Edixhoven entitled Diophantine Approximation and Abelian Varieties (2003).
Kálmán Győry, Debreceni Egyetem, Hungary
Kálmán Győry is Professor Emeritus at the University of Debrecen, a member of the Hungarian Academy of Sciences and a well-known researcher in Diophantine number theory. Over his career he has obtained several significant and pioneering results, among others on unit equations, decomposable form equations, and their various applications. His results have been published in one book and 160 research papers. Győry is also the founder and leader of the Number Theory Research Group in Debrecen, which consists of his former students and their descendants.