The Algorithmic Resolution of Diophantine Equations
The Algorithmic Resolution of Diophantine Equations
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.
- 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
Reviews & endorsements
"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 details
January 1999Paperback
9780521646338
260 pages
229 × 152 × 15 mm
0.39kg
Available
Table of Contents
- 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.
