Effective Results and Methods for Diophantine Equations over Finitely Generated Domains: (Series Number 475 London Mathematical Society Lecture Note Series)

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.

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.

About the Author :
Jan-Hendrik Evertse is Associate Professor in Number Theory at Leiden University in the Netherlands. He co-edited the lecture notes in mathematics Diophantine Approximation and Abelian Varieties (1993) with Bas Edixhoven, and co-authored two books with Kálmán Győry: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016). Kálmán Győry is Professor Emeritus at the University of Debrecen, Hungary and a member of the Hungarian Academy of Sciences. Győry is the founder and leader of the Number Theory Research Group in Debrecen. Together with Jan-Hendrik Evertse he has written two books: Unit Equations in Diophantine Number Theory (Cambridge, 2016) and Discriminant Equations in Diophantine Number Theory (Cambridge, 2016).

Review :
'… 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