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The Algorithmic Resolution of Diophantine Equations

The Algorithmic Resolution of Diophantine Equations
A Computational Cookbook


Part of London Mathematical Society Student Texts

  • Date Published: November 1998
  • availability: Available
  • format: Paperback
  • isbn: 9780521646338

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About the Authors
  • Beginning with a brief introduction to algorithms and diophantine equations, this volume aims to provide a coherent account of the methods used to find all the solutions to certain diophantine equations, particularly those procedures which have been developed for use on a computer. The study is divided into three parts, the emphasis throughout being on examining 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 which can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, mainly 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, with a basic knowledge of number theory, who are interested in solving diophantine equations using computational methods.

    • Introduction to an area of rapid growth
    • Gives readers sufficient grounding so that they can go on to read research literature in this subject
    • Suitable for graduate students of number theory
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    Reviews & endorsements

    '… should certainly establish itself as a key reference for established researchers and a natural starting point for new PhD students in the area.' E. V. Flynn, Bulletin of the London Mathematical Society

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    Product details

    • Date Published: November 1998
    • format: Paperback
    • isbn: 9780521646338
    • length: 260 pages
    • dimensions: 229 x 152 x 15 mm
    • weight: 0.39kg
    • availability: Available
  • Table of Contents

    1. Introduction
    Part I. Basic Solution Techniques:
    2. Local methods
    3. Applications of local methods to diophantine equations
    4. Ternary quadratic forms
    5. Computational diophantine approximation
    6. Applications of the LLL-algorithm
    Part II. Methods Using Linear Forms in Logarithms:
    7. Thue equations
    8. Thue–Mahler equations
    9. S-Unit equations
    10. Triangularly connected decomposable form equations
    11. Discriminant form equations
    Part III. Integral and Rational Points on Curves:
    12. Rational points on elliptic curves
    13. Integral points on elliptic curves
    14. Curves of genus greater than one

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    The Algorithmic Resolution of Diophantine Equations

    Nigel P. Smart

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    Nigel P. Smart, Hewlett-Packard Laboratories, Bristol

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