Effective Results and Methods for Diophantine Equations over Finitely Generated Domains
Effective Results and Methods for Diophantine Equations over Finitely Generated Domains
This book provides the first thorough treatment of effective results and methods for Diophantine equations over finitely generated domains. Compiling diverse results and techniques from papers written in recent decades, the text includes an in-depth analysis of classical equations including unit equations, Thue equations, hyper- and superelliptic equations, the Catalan equation, discriminant equations and decomposable form equations. The majority of results are proved in a quantitative form, giving effective bounds on the sizes of the solutions. The necessary techniques from Diophantine approximation and commutative algebra are all explained in detail without requiring any specialized knowledge on the topic, enabling readers from beginning graduate students to experts to prove effective finiteness results for various further classes of Diophantine equations.
- The first comprehensive treatment of effective results and methods for Diophantine equations over finitely generated characteristic 0 domains
- Provides an overview of results and techniques that were previously scattered across many papers
- Outlines all the necessary background material for beginning graduate students, including basic notions from algebraic number theory and the theory of algebraic function fields
Reviews & endorsements
'… I found the book to be presented and structured very well. It covers the topics and results that one would expect and hope to find in a book on this subject, as well as the new results mentioned above. But as the authors state towards the end of their preface, more possibilities exist for the application of their techniques. The authors have certainly done a good job of writing a clear, accessible account of this subject that should help to fulfill their hope that others will continue their work.' Paul M. Voutier, MathSciNet
Product details
No date availablePaperback
9781009005852
240 pages
230 × 152 × 13 mm
0.36kg
Table of Contents
- Preface
- Glossary of frequently used notation
- History and summary
- 1. Ineffective results for Diophantine equations over finitely generated domains
- 2. Effective results for Diophantine equations over finitely generated domains: the statements
- 3. A brief explanation of our effective methods over finitely generated domains
- 4. Effective results over number fields
- 5. Effective results over function fields
- 6. Tools from effective commutative algebra
- 7. The effective specialization method
- 8. Degree-height estimates
- 9. Proofs of the results from Sections 2.2–2.5 - use of specializations
- 10. Proofs of the results from Sections 2.6–2.8 - reduction to unit equations
- References
- Index.
