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Full Description
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.
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.
