The Algorithmic Resolution of Diophantine Equations
A Computational Cookbook

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

Nigel P. Smart (Author)

9780521641562, Cambridge University Press

Hardback, published 12 November 1998

260 pages
23.7 x 15.5 x 1.9 cm, 0.471 kg

'… 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

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.

Preface
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
Appendices
References
Index.

Subject Areas: Number theory [PBH]