Jan-Hendrik Evertse works in the Mathematical Institute at Leiden University. His research concentrates on Diophantine approximation and applications to Diophantine problems. In this area he has obtained some influential results, in particular on estimates for the numbers of solutions of Diophantine equations and inequalities. Kálmán Győry is Professor Emeritus at the University of Debrecen, a member of the Hungarian Academy of Sciences and a well-known researcher in Diophantine number theory. Over his career he has obtained several significant and pioneering results, among others on unit equations and decomposable form equations, and their various applications. Győry is also the founder and leader of the Number Theory Research Group in Debrecen, which consists of his former students and their descendants.

• Preface; Summary; Part I. Preliminaries: 1. Finite étale algebras over fields; 2. Dedekind domains; 3. Algebraic number fields; 4. Tools from the theory of unit equations; Part II. Monic Polynomials and Integral Elements of Given Discriminant, Monogenic Orders: 5. Basic finiteness theorems; 6. Effective results over Z; 7. Algorithmic resolution of discriminant form and index form equations; 8. Effective results over the S-integers of a number field; 9. The number of solutions of discriminant equations; 10. Effective results over finitely generated domains; 11. Further applications; Part III. Binary Forms of Given Discriminant: 12. A brief overview of the basic finiteness theorems; 13. Reduction theory of binary forms; 14. Effective results for binary forms of given discriminant; 15. Semi-effective results for binary forms of given discriminant; 16. Invariant orders of binary forms; 17. On the number of equivalence classes of binary forms of given discriminant; 18. Further applications; Glossary of frequently used notation; References; Index.

'… the book is very interesting and well written. It contains the motivational material necessary for those entering in the field of discriminant equations and succeeds to bring the reader to the forefront of research. Graduates and researchers in the field of number theory will find it a very valuable resource.' Dimitros Poulakis, Zentralblatt MATH