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A thorough introduction to the combinatorial approach to Mahler measure and more
Presents results that have not previously appeared in book form
Includes new tables of small Mahler measures and limit points
Appendices on prerequisites make the book selfcontained
Part of the book series: Universitext (UTX)
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Table of contents (22 chapters)

Front Matter
About this 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 twovariable 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 selfcontained, this textbook should also be accessible to wellprepared upperlevel undergraduates.
Keywords
 Mahler measure
 Lehmer's problem
 SchinzelZassenhaus conjecture
 Textbook on applications of graph theory to number theory
 adjacency matrix
 EstesGuralnick conjecture
 Robinson's conjecture
 Cyclotomic integers
 Auxiliary function methods
 SchurSiegelSmyth trace problem
 Symmetrizable integer matrics
 Signed and directed graphs
 Salem numbers
Reviews
Authors and Affiliations

Egham, UK
James McKee

Edinburgh, UK
Chris Smyth
About the authors
Chris Smyth, a professorial fellow in Number Theory at the University of Edinburgh, has a longstanding interest in Mahler measure. This dates from his PhD thesis, where he studied Lehmer’s conjecture for nonreciprocal integer polynomials. He discovered the first known closed formula for a 2dimensional Mahler measure involving an Lfunction, leading to a deep study of such formulae by Boyd, Deninger, Rodriguez Villegas and others. He invented the explicit auxiliary function method, which applies semiinfinite linear programming to numbertheoretic problems, including to the Mahler measure of totally real polynomials.
Bibliographic Information
Book Title: Around the Unit Circle
Book Subtitle: Mahler Measure, Integer Matrices and Roots of Unity
Authors: James McKee, Chris Smyth
Series Title: Universitext
DOI: https://doi.org/10.1007/9783030800314
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2021
Softcover ISBN: 9783030800307Published: 09 December 2021
eBook ISBN: 9783030800314Published: 08 December 2021
Series ISSN: 01725939
Series EISSN: 21916675
Edition Number: 1
Number of Pages: XX, 438
Number of Illustrations: 96 b/w illustrations, 2 illustrations in colour
Topics: Number Theory, Graph Theory, Linear Algebra