Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

Provides exceptional coverage of effective solutions for Diophantine equations over finitely generated domains.

Jan-Hendrik Evertse (Author), Kálmán Gy?ry (Author)

9781009005852, Cambridge University Press

Paperback / softback, published 28 April 2022

240 pages
23 x 15.2 x 1.3 cm, 0.36 kg

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.

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.

Subject Areas: Algebraic geometry [PBMW], Number theory [PBH], Algebra [PBF]