Number Theory
Number Theory

Author: Henri Cohen

Publisher: Springer Science & Business Media

Published: 2008-12-17

Total Pages: 619

ISBN-13: 038749894X

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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.

Number Theory
Number Theory

Author: Henri Cohen

Publisher: Springer Science & Business Media

Published: 2008-10-10

Total Pages: 673

ISBN-13: 0387499237

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The central theme of this book 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 book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

Diophantine Approximations and Diophantine Equations
Diophantine Approximations and Diophantine Equations

Author: Wolfgang M. Schmidt

Publisher: Springer

Published: 2006-12-08

Total Pages: 224

ISBN-13: 3540473742

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"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum

Arithmetic Geometry, Number Theory, and Computation
Arithmetic Geometry, Number Theory, and Computation

Author: Jennifer S. Balakrishnan

Publisher: Springer Nature

Published: 2022-03-15

Total Pages: 587

ISBN-13: 3030809145

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This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research. Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.

Number Theory
Number Theory

Author: Kağan Kurşungöz

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2021-11-08

Total Pages: 112

ISBN-13: 3110761114

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Three major branches of number theory are included in the volume: namely analytic number theory, algebraic number theory, and transcendental number theory. Original research is presented that discusses modern techniques and survey papers from selected academic scholars.

Diophantine Equations and Power Integral Bases
Diophantine Equations and Power Integral Bases

Author: István Gaál

Publisher: Springer Nature

Published: 2019-09-03

Total Pages: 335

ISBN-13: 3030238652

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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.