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Full Description
Discriminant equations are an important class of Diophantine equations with close ties to algebraic number theory, Diophantine approximation and Diophantine geometry. This book is the first comprehensive account of discriminant equations and their applications. It brings together many aspects, including effective results over number fields, effective results over finitely generated domains, estimates on the number of solutions, applications to algebraic integers of given discriminant, power integral bases, canonical number systems, root separation of polynomials and reduction of hyperelliptic curves. The authors' previous title, Unit Equations in Diophantine Number Theory, laid the groundwork by presenting important results that are used as tools in the present book. This material is briefly summarized in the introductory chapters along with the necessary basic algebra and algebraic number theory, making the book accessible to experts and young researchers alike.
Contents
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.
