Discriminant Equations in Diophantine Number Theory
Diophantine number theory is an active area that has seen tremendous progress over the past century. An important role in this theory is played by discriminant equations, a class of Diophantine equations with close ties to algebraic number theory, Diophantine approximation and Diophantine geometry. Discriminant equations are about univariate polynomials or binary forms of given discriminant, considered over various types of integral domains. This book is the irst comprehensive account of discriminant equations and their applications. It brings together many aspects, including effective results over number ields, effective results over initely generated domains, practical algorithms for solving concrete equations, estimates on the number of solutions, applications to algebraic integers of given discriminant, power integral bases, canonical number systems, algorithms for inding a minimal set of generators of an order of a number ield, 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 briely 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.