Cubic Forms and the Circle Method
Cubic Forms and the Circle Method

Author: Tim Browning

Publisher: Springer Nature

Published: 2021-11-19

Total Pages: 175

ISBN-13: 3030868729

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The Hardy–Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists. Not only is it capable of handling remarkably general systems of polynomial equations defined over arbitrary global fields, but it can also shed light on the space of rational curves that lie on algebraic varieties. This book, in which the arithmetic of cubic polynomials takes centre stage, is aimed at bringing beginning graduate students into contact with some of the many facets of the circle method, both classical and modern. This monograph is the winner of the 2021 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Algorithmic Number Theory
Algorithmic Number Theory

Author: Alf J. van der Poorten

Publisher: Springer

Published: 2008-05-07

Total Pages: 463

ISBN-13: 3540794565

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This book constitutes the refereed proceedings of the 8th International Algorithmic Number Theory Symposium, ANTS 2008, held in Banff, Canada, in May 2008. The 28 revised full papers presented together with 2 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on elliptic curves cryptology and generalizations, arithmetic of elliptic curves, integer factorization, K3 surfaces, number fields, point counting, arithmetic of function fields, modular forms, cryptography, and number theory.

Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms
Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms

Author: Wai Kiu Chan

Publisher: American Mathematical Soc.

Published: 2013

Total Pages: 259

ISBN-13: 0821883186

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This volume contains the proceedings of the International Workshop on Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms. The articles cover the arithmetic theory of quadratic forms and lattices, as well as the effective Diophantine analysis with height functions.

Combinatorial and Additive Number Theory III
Combinatorial and Additive Number Theory III

Author: Melvyn B. Nathanson

Publisher: Springer Nature

Published: 2019-12-10

Total Pages: 237

ISBN-13: 3030311066

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Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Geometry of Numbers
Geometry of Numbers

Author: C. G. Lekkerkerker

Publisher: Elsevier

Published: 2014-05-12

Total Pages: 521

ISBN-13: 1483259277

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Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume VIII: Geometry of Numbers focuses on bodies and lattices in the n-dimensional euclidean space. The text first discusses convex bodies and lattice points and the covering constant and inhomogeneous determinant of a set. Topics include the inhomogeneous determinant of a set, covering constant of a set, theorem of Minkowski-Hlawka, packing of convex bodies, successive minima and determinant of a set, successive minima of a convex body, extremal bodies, and polar reciprocal convex bodies. The publication ponders on star bodies, as well as points of critical lattices on the boundary, reducible, and irreducible star bodies and reduction of automorphic star bodies. The manuscript reviews homogeneous and inhomogeneous s forms and some methods. Discussions focus on asymmetric inequalities, inhomogeneous forms in more variables, indefinite binary quadratic forms, diophantine approximation, sums of powers of linear forms, spheres and quadratic forms, and a method of Blichfeldt and Mordell. The text is a dependable reference for researchers and mathematicians interested in bodies and lattices in the n-dimensional euclidean space.

History of the Theory of Numbers
History of the Theory of Numbers

Author: Leonard Eugene Dickson

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 324

ISBN-13: 9780821819364

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The last volume of Dickson's History is the most modern, covering quadratic and higher forms. The treatment here is more general than in Volume II, which, in a sense, is more concerned with special cases. Indeed, this volume chiefly presents methods of attacking whole classes of problems. Again, Dickson is exhaustive with references and citations.

History of the Theory of Numbers, Volume III
History of the Theory of Numbers, Volume III

Author: Leonard Eugene Dickson

Publisher: Courier Corporation

Published: 2005-06-03

Total Pages: 325

ISBN-13: 0486442349

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The three-volume series History of the Theory of Numbers is the work of the distinguished mathematician Leonard Eugene Dickson, who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory. This final volume in the series, which is suitable for upper-level undergraduates and graduate students, is devoted to quadratic and higher forms. It can be read independently of the preceding volumes, which explore divisibility and primality and diophantine analysis. Topics include reduction and equivalence of binary quadratic forms and representation of integers; composition of binary quadratic forms; the composition of orders and genera; irregular determinants; classes of binary quadratic forms with integral coefficients; binary quadratic forms whose coefficients are complete integers or integers of a field; classes of binary quadratic forms with complex integral coefficients; ternary and quaternary quadratic forms; cubic forms in three or more variables; binary hermitian forms; bilinear forms, matrices, and linear substitutions; congruencial theory of forms; and many other related topics. Indexes of authors cited and subjects appear at the end of the book.