Around the unit circle : Mahler measure, integer matrices and roots of unity / James McKee, Chris Smyth.
McKee, James (James Fraser), author.; Smyth, Chris, author.
2021
QA161.P59 M35 2021eb
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Title Around the unit circle : Mahler measure, integer matrices and roots of unity / James McKee, Chris Smyth.
Author McKee, James (James Fraser), author.
ISBN 9783030800314 (electronic bk.)
3030800318 (electronic bk.)
9783030800307
303080030X
DOI https://doi.org/10.1007/978-3-030-80031-4
Published Cham : Springer, [2021]
Copyright ©2021
Language English
Description 1 online resource (xx, 438 pages : illustrations (some color))
Item Number 10.1007/978-3-030-80031-4 doi
Call Number QA161.P59 M35 2021eb
Dewey Decimal Classification 512.9/422
Summary 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 Lehmers Problem (1933) and Boyds Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrovs proof of the SchinzelZassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinsons 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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed December 22, 2021).
Added Author Smyth, Chris, author.
Series Universitext.
Available in Other Form Around the unit circle.
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Mahler Measures of Polynomials in One Variable
Mahler Measures of Polynomials in Several Variables
Dobrowolski's Theorem
The Schinzel-Zassenhaus Conjecture
Roots of Unity and Cyclotomic Polynomials
Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem
Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem
The Set of Cassels Heights
Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tessellations
The Transfinite Diameter and Conjugate Sets of Algebraic Integers
Restricted Mahler Measure Results
The Mahler Measure of Nonreciprocal Polynomials
Minimal Noncyclotomic Integer Symmetric Matrices
The Method of Explicit Auxiliary Functions
The Method of Explicit Auxiliary Functions
Small-Span Integer Symmetric Matrices
Symmetrizable Matrices I: Introduction
Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices
Symmetrizable Matrices III: The Trace Problem
Salem Numbers from Graphs and Interlacing Quotients
Minimal Polynomials of Integer Symmetric Matrices
Breaking symmetry.

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