Discriminant Equations in Diophantine Number Theory 1st Edition by Jan Hendrik Evertse, Kálmán Győry 9781107097612 1107097614

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Product details:

ISBN 10: 1107097614

ISBN 13: 9781107097612

Author: Jan-Hendrik Evertse, Kálmán Győry

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.

Table of contents:

Part I. Preliminaries
Chapter 1: Finite étale algebras over fields
Chapter 2: Dedekind domains
Chapter 3: Algebraic number fields
Chapter 4: Tools from the theory of unit equations

Part II. Monic Polynomials and Integral Elements of Given Discriminant, Monogenic Orders
Chapter 5: Basic finiteness theorems
Chapter 6: Effective results over Z
Chapter 7: Algorithmic resolution of discriminant form and index form equations
Chapter 8: Effective results over the S-integers of a number field
Chapter 9: The number of solutions of discriminant equations
Chapter 10: Effective results over finitely generated domains
Chapter 11: Further applications

Part III. Binary Forms of Given Discriminant
Chapter 12: A brief overview of the basic finiteness theorems
Chapter 13: Reduction theory of binary forms
Chapter 14: Effective results for binary forms of given discriminant
Chapter 15: Semi-effective results for binary forms of given discriminant
Chapter 16: Invariant orders of binary forms
Chapter 17: On the number of equivalence classes of binary forms of given discriminant
Chapter 18: Further applications

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Tags: Jan Hendrik Evertse, Kálmán Győry, Discriminant, Equations