Elliptic Diophantine Equations
A Concrete Approach via the Elliptic Logarithm
Seiten
2013
De Gruyter (Verlag)
978-3-11-028091-3 (ISBN)
De Gruyter (Verlag)
978-3-11-028091-3 (ISBN)
This series is devoted to the publication of high-level monographs which cover the whole spectrum of current discrete mathematics and its applications in various fields. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of discrete mathematics.
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.
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.
Nikos Tzanakis, University of Crete, Heraklion, Greece.
| Erscheint lt. Verlag | 19.8.2013 |
|---|---|
| Reihe/Serie | De Gruyter Series in Discrete Mathematics and Applications ; 2 |
| Zusatzinfo | 2 b/w ill., 8 b/w tbl. |
| Verlagsort | Berlin/Boston |
| Sprache | englisch |
| Maße | 170 x 240 mm |
| Gewicht | 468 g |
| Themenwelt | Schulbuch / Wörterbuch ► Lexikon / Chroniken |
| Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika | |
| Mathematik / Informatik ► Mathematik ► Algebra | |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| Schlagworte | algebraic number theory • Computational Method • diophantine geometry • Elliptic Curves • Elliptic Diophantine Equation • Elliptic Diophantine Equation; Diophantine Geometry; Algebraic Number Theory; Magma; Computational Method • Elliptic Diophantine Equations • Elliptic Diophantine Equations; Diophantine Geometry; Algebraic Number Theory; Magma; Computational Methods and Techniques • Elliptische Gleichungen • Magma • Number Theory |
| ISBN-10 | 3-11-028091-4 / 3110280914 |
| ISBN-13 | 978-3-11-028091-3 / 9783110280913 |
| Zustand | Neuware |
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