Auxiliary Polynomials in Number Theory

Auxiliary Polynomials in Number Theory
Author :
Publisher : Cambridge University Press
Total Pages : 367
Release :
ISBN-10 : 9781316677636
ISBN-13 : 131667763X
Rating : 4/5 (36 Downloads)

Synopsis Auxiliary Polynomials in Number Theory by : David Masser

This unified account of various aspects of a powerful classical method, easy to understand in its simplest forms, is illustrated by applications in several areas of number theory. As well as including diophantine approximation and transcendence, which were mainly responsible for its invention, the author places the method in a broader context by exploring its application in other areas, such as exponential sums and counting problems in both finite fields and the field of rationals. Throughout the book, the method is explained in a 'molecular' fashion, where key ideas are introduced independently. Each application is the most elementary significant example of its kind and appears with detailed references to subsequent developments, making it accessible to advanced undergraduates as well as postgraduate students in number theory or related areas. It provides over 700 exercises both guiding and challenging, while the broad array of applications should interest professionals in fields from number theory to algebraic geometry.

Transcendental Number Theory

Transcendental Number Theory
Author :
Publisher : Cambridge University Press
Total Pages : 180
Release :
ISBN-10 : 052139791X
ISBN-13 : 9780521397919
Rating : 4/5 (1X Downloads)

Synopsis Transcendental Number Theory by : Alan Baker

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Number Theory and Polynomials

Number Theory and Polynomials
Author :
Publisher : Cambridge University Press
Total Pages : 350
Release :
ISBN-10 : 9780521714679
ISBN-13 : 0521714672
Rating : 4/5 (79 Downloads)

Synopsis Number Theory and Polynomials by : James Fraser McKee

Contributions by leading experts in the field provide a snapshot of current progress in polynomials and number theory.

Around the Unit Circle

Around the Unit Circle
Author :
Publisher : Springer Nature
Total Pages : 444
Release :
ISBN-10 : 9783030800314
ISBN-13 : 3030800318
Rating : 4/5 (14 Downloads)

Synopsis Around the Unit Circle by : James McKee

Mahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book. One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some for computer programming. A wide range of stimulating open problems is also included. At the end of each chapter there is a glossary of newly introduced concepts and definitions. Around the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates.

Number Theory with Computations

Number Theory with Computations
Author :
Publisher : Springer Nature
Total Pages : 445
Release :
ISBN-10 : 9783031638145
ISBN-13 : 303163814X
Rating : 4/5 (45 Downloads)

Synopsis Number Theory with Computations by : Peter Shiu

Analytic Number Theory

Analytic Number Theory
Author :
Publisher : Cambridge University Press
Total Pages : 396
Release :
ISBN-10 : 9780521625128
ISBN-13 : 0521625122
Rating : 4/5 (28 Downloads)

Synopsis Analytic Number Theory by : Yoichi Motohashi

Authoritative, up-to-date review of analytic number theory containing outstanding contributions from leading international figures.

Analytic Number Theory

Analytic Number Theory
Author :
Publisher : Springer Science & Business Media
Total Pages : 557
Release :
ISBN-10 : 9781461234647
ISBN-13 : 1461234646
Rating : 4/5 (47 Downloads)

Synopsis Analytic Number Theory by : B. Berndt

On April 25-27, 1989, over a hundred mathematicians, including eleven from abroad, gathered at the University of Illinois Conference Center at Allerton Park for a major conference on analytic number theory. The occa sion marked the seventieth birthday and impending (official) retirement of Paul T. Bateman, a prominent number theorist and member of the mathe matics faculty at the University of Illinois for almost forty years. For fifteen of these years, he served as head of the mathematics department. The conference featured a total of fifty-four talks, including ten in vited lectures by H. Delange, P. Erdos, H. Iwaniec, M. Knopp, M. Mendes France, H. L. Montgomery, C. Pomerance, W. Schmidt, H. Stark, and R. C. Vaughan. This volume represents the contents of thirty of these talks as well as two further contributions. The papers span a wide range of topics in number theory, with a majority in analytic number theory.

Polynomial Methods in Combinatorics

Polynomial Methods in Combinatorics
Author :
Publisher : American Mathematical Soc.
Total Pages : 287
Release :
ISBN-10 : 9781470428907
ISBN-13 : 1470428903
Rating : 4/5 (07 Downloads)

Synopsis Polynomial Methods in Combinatorics by : Larry Guth

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

Number Theory in Progress

Number Theory in Progress
Author :
Publisher : Walter de Gruyter
Total Pages : 1212
Release :
ISBN-10 : 9783110285581
ISBN-13 : 3110285584
Rating : 4/5 (81 Downloads)

Synopsis Number Theory in Progress by : Kálmán Györy

Proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the 60th Birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997.

Unit Equations in Diophantine Number Theory

Unit Equations in Diophantine Number Theory
Author :
Publisher : Cambridge University Press
Total Pages : 381
Release :
ISBN-10 : 9781316432358
ISBN-13 : 1316432351
Rating : 4/5 (58 Downloads)

Synopsis Unit Equations in Diophantine Number Theory by : Jan-Hendrik Evertse

Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.