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The 13 chapters of this book centre around the proof of Theorem 1
of Faltings' paper "Diophantine approximation on abelian
varieties," Ann. Math.133 (1991) and together give an approach to
the proof that is accessible to Ph.D-level students in number
theory and algebraic geometry. Each chapter is based on an
instructional lecture given by its author ata special conference
for graduate students, on the topic of Faltings' paper.
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.
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.
Diophantine number theory is an active area that has seen
tremendous growth over the past century, and in this theory unit
equations play a central role. This comprehensive treatment is the
first volume devoted to these equations. The authors gather
together all the most important results and look at many different
aspects, including effective results on unit equations over number
fields, estimates on the number of solutions, analogues for
function fields and effective results for unit equations over
finitely generated domains. They also present a variety of
applications. Introductory chapters provide the necessary
background in algebraic number theory and function field theory, as
well as an account of the required tools from Diophantine
approximation and transcendence theory. This makes the book
suitable for young researchers as well as experts who are looking
for an up-to-date overview of the field.
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