The Algorithmic Resolution of Diophantine Equations: A Computational Cookbook (London Mathematical Society Student Texts)

The Algorithmic Resolution of Diophantine Equations: A Computational Cookbook (London Mathematical Society Student Texts)
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Nigel P. Smart

Review
"It is high time for such a book to appear...professional mathematicians and even experts in the subject will find it useful as well...Smart did a good job,and his book will efficiently serve as a textbook for beginners and as a reference source for the experts." Mathematical Reviews

"It is a real pleasure to read this book, mainly because the author gives many examples and many practical remarks concerning the effective solution of diophantine equations. ...this is a very attractive book, full of concrete information, which gives a very clear and lucid view of the current knowledge." Bulletin of the AMS


Product Description
Beginning with a brief introduction to algorithms and diophantine equations, this volume provides a coherent modern account of the methods used to find all the solutions to certain diophantine equations, particularly those developed for use on a computer. The study is divided into three parts, emphasizing approaches with a wide range of applications. The first section considers basic techniques including local methods, sieving, descent arguments and the LLL algorithm. The second section explores problems that can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, focusing on rational and integral points on elliptic curves. Each chapter concludes with a useful set of exercises. A detailed bibliography is included. This book will appeal to graduate students and research workers interested in solving diophantine equations using computational methods.
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Product Details

* Hardcover: 260 pages
* Publisher: Cambridge University Press (January 13, 1999)
* Language: English
* ISBN-10: 052164156X
* ISBN-13: 978-0521641562

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The Decomposition of Primes in Torsion Point Fields (Lecture Notes in Mathematics)

The Decomposition of Primes in Torsion Point Fields (Lecture Notes in Mathematics)
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Adelmann C.

Product Description
This text deals in nonabelian extensions of number fields. The necessary prerequisites from the theory of elliptic curves, modular forms, algebraic number theory, and invariant theory are explained. Softcover.
Product Details

* Paperback: 147 pages
* Publisher: Springer; 1 edition (June 15, 2001)
* Language: English
* ISBN-10: 3540420355
* ISBN-13: 978-3540420354

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Analytic Methods for Diophantine Equations and Diophantine Inequalities (Cambridge Mathematical Library)

Analytic Methods for Diophantine Equations and Diophantine Inequalities (Cambridge Mathematical Library)
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H. Davenport (Author), T. D. Browning (Author)

Review
"Analytic Methods stands on its own as yet another great example of Davenport's style and ability to present a number theory topic. Moreover, the new edition of his lecture notes include a foreword written by three experts...where the recent discoveries and state of the art on the topics covered in the book are summarized, adding a great amount to the total value of the volume."
MAA Reviews, Alvara Lozano-Robledo, Cornell University


Product Description
Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today as it was when the book originally appeared. The three main themes of the book are Waring's problem and the representation of integers by diagonal forms, the solubility in integers of systems of forms in many variables, and the solubility in integers of diagonal inequalities. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added.
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Product Details

* Paperback: 160 pages
* Publisher: Cambridge University Press; 2 edition (February 7, 2005)
* Language: English
* ISBN-10: 0521605830
* ISBN-13: 978-0521605830

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Diophantine Approximations and Diophantine Equations (Lecture Notes in Mathematics)

Diophantine Approximations and Diophantine Equations (Lecture Notes in Mathematics)
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Wolfgang M. Schmidt

Review
"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum


Product Description
"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum
Product Details

* Paperback: 217 pages
* Publisher: Springer (June 9, 2000)
* Language: English
* ISBN-10: 354054058X
* ISBN-13: 978-3540540588

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Trigonometric sums in number theory and analysis

Trigonometric sums in number theory and analysis
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Arkhipov G., Chubarikov V., Karatsuba A.

Product Description
The book presents the theory of multiple trigonometric sums constructed by the authors. Following a unified approach, the authors obtain estimates for these sums similar to the classical I. M. Vinogradov's estimates and use them to solve several problems in analytic number theory. They investigate trigonometric integrals, which are often encountered in physics, mathematical statistics, and analysis, and present purely arithmetic results concerning the solvability of equations in integers.
Product Details

* Hardcover: 554 pages
* Publisher: Walter de Gruyter (November 30, 2004)
* Language: English
* ISBN-10: 3110162660
* ISBN-13: 978-3110162660

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Logarithmic Forms and Diophantine Geometry (New Mathematical Monographs)

Logarithmic Forms and Diophantine Geometry (New Mathematical Monographs)
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A. Baker (Author), G. Wüstholz (Author)

Review
"This book gives the necessary intuitive background to study the original journal articles of Baker, Masser, Wüstholz and others..."
Yuri Bilu, Mathematical Reviews


Product Description
There is now much interplay between studies on logarithmic forms and deep aspects of arithmetic algebraic geometry. New light has been shed, for instance, on the famous conjectures of Tate and Shafarevich relating to abelian varieties and the associated celebrated discoveries of Faltings establishing the Mordell conjecture. This book gives an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years. The first part covers basic material in transcendental number theory but with a modern perspective. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, several advanced topics. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the André-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in Diophantine geometry.
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Product Details

* Hardcover: 208 pages
* Publisher: Cambridge University Press; 1 edition (February 18, 2008)
* Language: English
* ISBN-10: 0521882680
* ISBN-13: 978-0521882682

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Analytic Number Theory: An Introductory Course

Analytic Number Theory: An Introductory Course
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Bateman P., Diamond H.

Review
...suitable for beginning graduate students, or possibly even advanced undergraduates. -- Zentralblatt MATH

A nice well-written book with plenty of material for a one-year graduate course. -- MAA Online Book Review

…a nice introductory book on analytic number theory for students or readers with some background. -- Mathematical Reviews --This text refers to the Hardcover edition.


Product Description
This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable ("elementary") and complex variable ("analytic") methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed.
Product Details

* Paperback: 376 pages
* Publisher: World Scientific Publishing Company (September 30, 2004)
* Language: English
* ISBN-10: 9812560807
* ISBN-13: 978-9812560803

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