Diophantine Equations Over Function Fields
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Diophantine Equations Over Function Fields
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Author : R. C. Mason
language : en
Publisher: Cambridge University Press
Release Date : 1984-04-26
Diophantine Equations Over Function Fields written by R. C. Mason and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-04-26 with Mathematics categories.
A self-contained account of a new approach to the subject.
Diophantine Equations Over Function Fields
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Author : R. C. Mason
language : en
Publisher: Cambridge University Press
Release Date : 1984-04-26
Diophantine Equations Over Function Fields written by R. C. Mason and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-04-26 with Mathematics categories.
Diophantine equations over number fields have formed one of the most important and fruitful areas of mathematics throughout civilisation. In recent years increasing interest has been aroused in the analogous area of equations over function fields. However, although considerable progress has been made by previous authors, none has attempted the central problem of providing methods for the actual solution of such equations. The latter is the purpose and achievement of this volume: algorithms are provided for the complete resolution of various families of equations, such as those of Thue, hyperelliptic and genus one type. The results are achieved by means of an original fundamental inequality, first announced by the author in 1982. Several specific examples are included as illustrations of the general method and as a testimony to its efficiency. Furthermore, bounds are obtained on the solutions which improve on those obtained previously by other means. Extending the equality to a different setting, namely that of positive characteristic, enables the various families of equations to be resolved in that circumstance. Finally, by applying the inequality in a different manner, simple bounds are determined on their solutions in rational functions of the general superelliptic equation. This book represents a self-contained account of a new approach to the subject, and one which plainly has not reached the full extent of its application. It also provides a more direct on the problems than any previous book. Little expert knowledge is required to follow the theory presented, and it will appeal to professional mathematicians, research students and the enthusiastic undergraduate.
Unit Equations In Diophantine Number Theory
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Author : Jan-Hendrik Evertse
language : en
Publisher: Cambridge University Press
Release Date : 2015-12-30
Unit Equations In Diophantine Number Theory written by Jan-Hendrik Evertse and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-12-30 with Mathematics categories.
A comprehensive, graduate-level treatment of unit equations and their various applications.
Lecture Notes On Diophantine Analysis
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Author : Umberto Zannier
language : en
Publisher: Springer
Release Date : 2015-05-05
Lecture Notes On Diophantine Analysis written by Umberto Zannier and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-05 with Mathematics categories.
These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables. This case historically suggested some major ideas for more general problems. Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail. In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation. The book also contains several supplements, hinted exercises and an appendix on recent work on heights.
Function Field Arithmetic
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Author : Dinesh S. Thakur
language : en
Publisher: World Scientific
Release Date : 2004
Function Field Arithmetic written by Dinesh S. Thakur and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
This book provides an exposition of function field arithmetic with emphasis on recent developments concerning Drinfeld modules, the arithmetic of special values of transcendental functions (such as zeta and gamma functions and their interpolations), diophantine approximation and related interesting open problems. While it covers many topics treated in 'Basic Structures of Function Field Arithmetic' by David Goss, it complements that book with the inclusion of recent developments as well as the treatment of new topics such as diophantine approximation, hypergeometric functions, modular forms, transcendence, automata and solitons. There is also new work on multizeta values and log-algebraicity. The author has included numerous worked-out examples. Many open problems, which can serve as good thesis problems, are discussed.
Equations Over Finite Fields
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Author : W.M. Schmidt
language : en
Publisher: Springer
Release Date : 2006-11-14
Equations Over Finite Fields written by W.M. Schmidt and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
Diophantine Analysis
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Author : Jörn Steuding
language : en
Publisher: Birkhäuser
Release Date : 2016-12-21
Diophantine Analysis written by Jörn Steuding and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-12-21 with Mathematics categories.
This collection of course notes from a number theory summer school focus on aspects of Diophantine Analysis, addressed to Master and doctoral students as well as everyone who wants to learn the subject. The topics range from Baker’s method of bounding linear forms in logarithms (authored by Sanda Bujačić and Alan Filipin), metric diophantine approximation discussing in particular the yet unsolved Littlewood conjecture (by Simon Kristensen), Minkowski’s geometry of numbers and modern variations by Bombieri and Schmidt (Tapani Matala-aho), and a historical account of related number theory(ists) at the turn of the 19th Century (Nicola M.R. Oswald). Each of these notes serves as an essentially self-contained introduction to the topic. The reader gets a thorough impression of Diophantine Analysis by its central results, relevant applications and open problems. The notes are complemented with many references and an extensive register which makes it easy to navigate through the book.
Classical Diophantine Equations
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Author : Vladimir G. Sprindzuk
language : en
Publisher: Springer
Release Date : 2006-11-15
Classical Diophantine Equations written by Vladimir G. Sprindzuk and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, now that the book appears in English, close studyand emulation. In particular those emphases allow him to devote the eighth chapter to an analysis of the interrelationship of the class number of algebraic number fields involved and the bounds on the heights of thesolutions of the diophantine equations. Those ideas warrant further development. The final chapter deals with effective aspects of the Hilbert Irreducibility Theorem, harkening back to earlier work of the author. There is no other congenial entry point to the ideas of the last two chapters in the literature.
Solving The Pell Equation
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Author : Michael Jacobson
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-02
Solving The Pell Equation written by Michael Jacobson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-02 with Mathematics categories.
Pell’s Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell’s Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell’s Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.
Equations Over Finite Fields
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Author : Wolfgang M. Schmidt
language : en
Publisher:
Release Date : 2004
Equations Over Finite Fields written by Wolfgang M. Schmidt and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.