The Algorithmic Resolution Of Diophantine Equations

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The Algorithmic Resolution Of Diophantine Equations

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Author : Nigel P. Smart
language : en
Publisher: Cambridge University Press
Release Date : 1998-11-12

The Algorithmic Resolution Of Diophantine Equations written by Nigel P. Smart and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-11-12 with Mathematics categories.

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.

The Algorithmic Resolution Of Diophantine Equations

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Author : Nigel P. Smart
language : en
Publisher: Cambridge University Press
Release Date : 1998-11-12

The Algorithmic Resolution Of Diophantine Equations written by Nigel P. Smart and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-11-12 with Mathematics categories.

A coherent account of the computational methods used to solve diophantine equations.

The Algorithmic Resolution Of Diophantine Equations

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Author : Nigel Paul Smart
language : en
Publisher:
Release Date : 2014-05-14

The Algorithmic Resolution Of Diophantine Equations written by Nigel Paul Smart and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-14 with MATHEMATICS categories.

A coherent account of the computational methods used to solve diophantine equations.

The Algorithmic Resolution Of Diophantine Equations

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Author : Nigel P. Smart
language : en
Publisher:
Release Date : 1998

The Algorithmic Resolution Of Diophantine Equations written by Nigel P. Smart and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Diophantine equations categories.

A coherent account of the computational methods used to solve diophantine equations.

Diophantine Equations And Power Integral Bases

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Author : István Gaál
language : en
Publisher: Springer Nature
Release Date : 2019-09-03

Diophantine Equations And Power Integral Bases written by István Gaál and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-03 with Mathematics categories.

Work examines the latest algorithms and tools to solve classical types of diophantine equations.; Unique book---closest competitor, Smart, Cambridge, does not treat index form equations.; Author is a leading researcher in the field of computational algebraic number theory.; The text is illustrated with several tables of various number fields, including their data on power integral bases.; Several interesting properties of number fields are examined.; Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations.

Polynomial Diophantine Equations

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Author : Bogdan Grechuk
language : en
Publisher: Springer Nature
Release Date : 2024-09-01

Polynomial Diophantine Equations written by Bogdan Grechuk and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-09-01 with Mathematics categories.

This book proposes a novel approach to the study of Diophantine equations: define an appropriate version of the equation’s size, order all polynomial Diophantine equations by size, and then solve the equations in order. Natural questions about the solution set of Diophantine equations are studied in this book using this approach. Is the set empty? Is it finite or infinite? Can all integer solutions be parametrized? By ordering equations by size, the book attempts to answer these questions in a systematic manner. When the size grows, the difficulty of finding solutions increases and the methods required to determine solutions become more advanced. Along the way, the reader will learn dozens of methods for solving Diophantine equations, each of which is illustrated by worked examples and exercises. The book ends with solutions to exercises and a large collection of open problems, often simple to write down yet still unsolved. The original approach pursued in this book makes it widely accessible. Many equations require only high school mathematics and creativity to be solved, so a large part of the book is accessible to high school students, especially those interested in mathematical competitions such as olympiads. The main intended audience is undergraduate students, for whom the book will serve as an unusually rich introduction to the topic of Diophantine equations. Many methods from the book will be useful for graduate students, while Ph.D. students and researchers may use it as a source of fascinating open questions of varying levels of difficulty.

Diophantine Equations And Power Integral Bases

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Author : Istvan Gaal
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Diophantine Equations And Power Integral Bases written by Istvan Gaal and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

This monograph investigates algorithms for determining power integral bases in algebraic number fields. It introduces the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms. Particular emphasis is placed on properties of number fields and new applications. The text is illustrated with several tables of various number fields, including their data on power integral bases. Good resource for solving classical types of diophantine equations. Aimed at advanced undergraduate/graduate students and researchers.

Elliptic Diophantine Equations

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Author : Nikos Tzanakis
language : en
Publisher: Walter de Gruyter
Release Date : 2013-08-29

Elliptic Diophantine Equations written by Nikos Tzanakis and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-08-29 with Mathematics categories.

This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.

Effective Results And Methods For Diophantine Equations Over Finitely Generated Domains

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Author : Jan-Hendrik Evertse
language : en
Publisher: Cambridge University Press
Release Date : 2022-04-28

Effective Results And Methods For Diophantine Equations Over Finitely Generated Domains written by Jan-Hendrik Evertse and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-28 with Mathematics categories.

Provides exceptional coverage of effective solutions for Diophantine equations over finitely generated domains.

Unit Equations In Diophantine Number Theory

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Author : Jan-Hendrik Evertse
language : en
Publisher: Cambridge University Press
Release Date : 2015-12-30

Unit Equations In Diophantine Number Theory written by Jan-Hendrik Evertse and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-12-30 with Mathematics categories.

A comprehensive, graduate-level treatment of unit equations and their various applications.