Elliptic diophantine equations / by Nikos Tzanakis.

Format:

Book

Author/Creator:

Tzanakis, Nikos, 1952-

Series:

De Gruyter Series in Discrete Mathematics and Applications

De Gruyter Series in Discrete Mathematics and Applications ; 2

Language:

English

Subjects (All):

Diophantine equations.

Elliptic functions.

Physical Description:

1 online resource (196 p.)

Edition:

1st ed.

Place of Publication:

Berlin ; Boston : Walter de Gruyter, [2013]

Language Note:

English

Summary:

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.

Contents:

Front matter

Preface

Contents

Chapter 1 Elliptic curves and equations

Chapter 2 Heights

Chapter 3 Weierstrass equations over C and R

Chapter 4 The elliptic logarithm method

Chapter 5 Linear form for the Weierstrass equation

Chapter 6 Linear form for the quartic equation

Chapter 7 Linear form for simultaneous Pell equations

Chapter 8 Linear form for the general elliptic equation

Chapter 9 Bound for the coefficients of the linear form

Chapter 10 Reducing the bound obtained in Chapter 9

Chapter 11 S-integer solutions of Weierstrass equations

List of symbols

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9783110281149

3110281147

OCLC:

858762166