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.

‘… 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 Source: MathSciNet