Polynomial Diophantine Equations
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Polynomial Diophantine Equations
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Author : Bogdan Grechuk
language : en
Publisher: Springer Nature
Release Date : 2024-09-01
Polynomial Diophantine Equations written by Bogdan Grechuk and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-09-01 with Mathematics categories.
This book proposes a novel approach to the study of Diophantine equations: define an appropriate version of the equation’s size, order all polynomial Diophantine equations by size, and then solve the equations in order. Natural questions about the solution set of Diophantine equations are studied in this book using this approach. Is the set empty? Is it finite or infinite? Can all integer solutions be parametrized? By ordering equations by size, the book attempts to answer these questions in a systematic manner. When the size grows, the difficulty of finding solutions increases and the methods required to determine solutions become more advanced. Along the way, the reader will learn dozens of methods for solving Diophantine equations, each of which is illustrated by worked examples and exercises. The book ends with solutions to exercises and a large collection of open problems, often simple to write down yet still unsolved. The original approach pursued in this book makes it widely accessible. Many equations require only high school mathematics and creativity to be solved, so a large part of the book is accessible to high school students, especially those interested in mathematical competitions such as olympiads. The main intended audience is undergraduate students, for whom the book will serve as an unusually rich introduction to the topic of Diophantine equations. Many methods from the book will be useful for graduate students, while Ph.D. students and researchers may use it as a source of fascinating open questions of varying levels of difficulty.
Elliptic Tales
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Author : Avner Ash
language : en
Publisher: Princeton University Press
Release Date : 2012
Elliptic Tales written by Avner Ash and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
Describes the latest developments in number theory by looking at the Birch and Swinnerton-Dyer Conjecture.
Polynomial Diophantine Equations
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Author : Bogdan Grechuk
language : en
Publisher:
Release Date : 2024
Polynomial Diophantine Equations written by Bogdan Grechuk and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024 with Number theory categories.
1 Explicit Solutions to Small Equations -- 2 Existence of Polynomial Parametrizations -- 3 Equations in Two Variables -- 4 Explicit Descriptions of the Solution Sets -- 5 The Finiteness Problem -- 6 Existence of Non-Trivial Solutions to Homogeneous Equations -- 7 Existence of Integer Solutions -- 8 Alternative Orderings of Diophantine Equations -- 9 Solutions to Selected Exercises (by Ashleigh Wilcox).
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.
Selecta Diophantine Problems And Polynomials
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Author : Andrzej Schinzel
language : en
Publisher: European Mathematical Society
Release Date : 2007
Selecta Diophantine Problems And Polynomials written by Andrzej Schinzel and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Analyse diophantienne categories.
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.
A Troupe Of Special Second Degree Multivariable Polynomial Diophantine Equations With Integer Solutions
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Author : M. A. Gopalan
language : en
Publisher: Deep Science Publishing
Release Date : 2025-01-24
A Troupe Of Special Second Degree Multivariable Polynomial Diophantine Equations With Integer Solutions written by M. A. Gopalan and has been published by Deep Science Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-24 with Mathematics categories.
The fascinating branch of Mathematics is the Theory of Numbers in which the subject of Diophantine equations requiring only the integer solutions is an interesting area to various mathematicians and to the lovers of mathematics because it is a treasure house in which the search for many hidden connections is a treasure hunt. In other words, the theory of Diophantine equations is an ancient subject that typically involves solving, polynomial equation in two or more variables or a system of polynomial equations with the number of unknowns greater than the number of equations, in integers and occupies a pivotal role in the region of mathematics. It is worth to mention that the Diophantine problems are plenty playing a significant role in the development of mathematics because the beauty of Diophantine equations is that the number of equations is less than the number of unknowns. The theory of Diophantine equations provides a fertile ground for both professionals and amateurs. In addition to known results, the theory of Diophantine equations abounds with unsolved problems (Carmichael.,1959; Dickson.,1952; Mordell.,1969). In this context, for simplicity and brevity, one may refer Gopalan et.al., 2012, 2015, 2021, 2024; Mahalakshmi, Shanthi., 2023; Sathiyapriya et.al., 2024; Shanthi.,2023; Shanthi, Mahalakshmi.,2023; Shanthi, Gopalan.,2024; Thiruniraiselvi, Gopalan., 2024; Vidhyalakshmi et.al., 2022 for some binary and ternary quadratic Diophantine equations. Although many of its results can be stated in simple and elegant terms, their proofs are sometimes long and complicated. Many unsolved problems that have been daunting mathematicians for centuries provide unlimited opportunities to expand the frontiers of mathematical knowledge. The subject of Diophantine equations has fascinated and inspired both amateurs and mathematicians alike and so they merit special recognition. The successful completion of exhibiting all integers satisfying the requirements set forth in the problem add to further progress of Number Theory as they offer good applications in the field of Graph theory, Modular theory, Coding and Cryptography, Engineering, Music and so on. Integers have repeatedly played a crucial role in the evolution of the Natural Sciences. The theory of integers provides answers to real world problems. The focus in this book is on solving multivariable second-degree Diophantine equations. These types of equations can be challenging as they involve finding integer solutions that satisfy the given polynomial equation. Learning about the various techniques to solve these multivariable second-degree Diophantine equations in successfully deriving their solutions help us understand how numbers work and their significance in different areas of mathematics and science. This book contains a reasonable collection of special multivariable Quadratic Diophantine problems in three, four and five variables. The process of getting different sets of integer solutions to each of the quadratic Diophantine equations considered in this book are illustrated in an elegant manner.
Exponential Diophantine Equations
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Author : T. N. Shorey
language : en
Publisher: Cambridge University Press
Release Date : 1986-11-27
Exponential Diophantine Equations written by T. N. Shorey 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 1986-11-27 with Mathematics categories.
This is a integrated presentation of the theory of exponential diophantine equations. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker theory of linear forms in logarithms of algebraic numbers. Topics covered include the Thue equations, the generalised hyperelliptic equation, and the Fermat and Catalan equations. The necessary preliminaries are given in the first three chapters. Each chapter ends with a section giving details of related results.
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.
Effective Polynomial Computation
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Author : Richard Zippel
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Effective Polynomial Computation written by Richard Zippel 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 Computers categories.
Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth. Preparatory to the discussion of algorithms for polynomials, the first third of this book discusses related issues in elementary number theory. These results are either used in later algorithms (e.g. the discussion of lattices and Diophantine approximation), or analogs of the number theoretic algorithms are used for polynomial problems (e.g. Euclidean algorithm and p-adic numbers). Among the unique features of Effective Polynomial Computation is the detailed material on greatest common divisor and factoring algorithms for sparse multivariate polynomials. In addition, both deterministic and probabilistic algorithms for irreducibility testing of polynomials are discussed.