Diophantine Analysis And Related Fields

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Diophantine Analysis

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Author : Jorn Steuding
language : en
Publisher: CRC Press
Release Date : 2005-05-19

Diophantine Analysis written by Jorn Steuding and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-05-19 with Mathematics categories.

While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications. He also explores modern topics rarely addressed in other texts, including the abc conjecture, the polynomial Pell equation, and the irrationality of the zeta function and touches on topics and applications related to discrete mathematics, such as factoring methods for large integers. Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field. A detailed appendix supplies the necessary background material, more than 200 exercises reinforce the concepts, and engaging historical notes bring the subject to life.

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.

Diophantine Analysis And Related Fields 2011 Darf 2011

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Author : Amou Masaaki
language : en
Publisher: American Institute of Physics
Release Date : 2011-10-04

Diophantine Analysis And Related Fields 2011 Darf 2011 written by Amou Masaaki and has been published by American Institute of Physics this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-04 with Science categories.

The book covers the topics mainly from analytic number theory; its scope ranges from Diophantine approximations and equations, together with their applications, and transcendence theory to various zeta functions. The book is especially useful for the front researchers and graduate students who are interested in number theory and its relevant areas.

Diophantine Analysis And Related Fields 2010

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Author : Takao Komatsu
language : en
Publisher: American Institute of Physics
Release Date : 2010-08-13

Diophantine Analysis And Related Fields 2010 written by Takao Komatsu and has been published by American Institute of Physics this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-08-13 with Mathematics categories.

The purpose of the conference was to report recent progress and developments of diophantine aspects of analytic number theory, especially focusing upon the topics in diophantine analysis and related fields; its simultaneous objectives are to promote interactions between analytic number theorists and mathematicians from other fields, and further to create opportunities for opening up new individual/collaborative research projects.

Fundamentals Of Diophantine Geometry

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Author : S. Lang
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29

Fundamentals Of Diophantine Geometry written by S. Lang 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 2013-06-29 with Mathematics categories.

Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.

Elliptic Curves

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Author : S. Lang
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29

Elliptic Curves written by S. Lang 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 2013-06-29 with Mathematics categories.

It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.

Diophantine Equations And Power Integral Bases

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Author : Istvan Gaal
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Diophantine Equations And Power Integral Bases written by Istvan Gaal 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 2012-12-06 with Mathematics categories.

This monograph investigates algorithms for determining power integral bases in algebraic number fields. It introduces the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms. Particular emphasis is placed on properties of number fields and new applications. The text is illustrated with several tables of various number fields, including their data on power integral bases. Good resource for solving classical types of diophantine equations. Aimed at advanced undergraduate/graduate students and researchers.

Diophantine Equations And Power Integral Bases

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Author : István Gaál
language : en
Publisher: Springer Nature
Release Date : 2019-09-03

Diophantine Equations And Power Integral Bases written by István Gaál and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-03 with Mathematics categories.

Work examines the latest algorithms and tools to solve classical types of diophantine equations.; Unique book---closest competitor, Smart, Cambridge, does not treat index form equations.; Author is a leading researcher in the field of computational algebraic number theory.; The text is illustrated with several tables of various number fields, including their data on power integral bases.; Several interesting properties of number fields are examined.; Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations.

Diophantine Analysis And Related Fields 2006

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Author : Masanori Katsurada
language : en
Publisher:
Release Date : 2006

Diophantine Analysis And Related Fields 2006 written by Masanori Katsurada and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with categories.

Number Theory

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Author : Henri Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-17

Number Theory written by Henri Cohen 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 2008-12-17 with Mathematics categories.

This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.