Skip to main content
  • Textbook
  • © 2021

Around the Unit Circle

Mahler Measure, Integer Matrices and Roots of Unity

Authors:

(view affiliations)
  • 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 self-contained

Part of the book series: Universitext (UTX)

Buying options

eBook
USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-80031-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 79.99
Price excludes VAT (USA)

This is a preview of subscription content, access via your institution.

Table of contents (22 chapters)

  1. Front Matter

    Pages i-xx
  2. Mahler Measures of Polynomials in One Variable

    • James McKee, Chris Smyth
    Pages 1-24
  3. Mahler Measures of Polynomials in Several Variables

    • James McKee, Chris Smyth
    Pages 25-56
  4. Dobrowolski’s Theorem

    • James McKee, Chris Smyth
    Pages 57-67
  5. The Schinzel–Zassenhaus Conjecture

    • James McKee, Chris Smyth
    Pages 69-75
  6. Roots of Unity and Cyclotomic Polynomials

    • James McKee, Chris Smyth
    Pages 77-112
  7. The Set of Cassels Heights

    • James McKee, Chris Smyth
    Pages 149-169
  8. Restricted Mahler Measure Results

    • James McKee, Chris Smyth
    Pages 205-218
  9. The Mahler Measure of Nonreciprocal Polynomials

    • James McKee, Chris Smyth
    Pages 219-236
  10. Minimal Noncyclotomic Integer Symmetric Matrices

    • James McKee, Chris Smyth
    Pages 237-248
  11. The Method of Explicit Auxiliary Functions

    • James McKee, Chris Smyth
    Pages 249-268
  12. The Trace Problem for Integer Symmetric Matrices

    • James McKee, Chris Smyth
    Pages 269-276
  13. Small-Span Integer Symmetric Matrices

    • James McKee, Chris Smyth
    Pages 277-300
  14. Symmetrizable Matrices I: Introduction

    • James McKee, Chris Smyth
    Pages 301-315
  15. Symmetrizable Matrices III: The Trace Problem

    • James McKee, Chris Smyth
    Pages 333-342

About this book

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.

Keywords

  • Mahler measure
  • Lehmer's problem
  • Schinzel-Zassenhaus conjecture
  • Textbook on applications of graph theory to number theory
  • adjacency matrix
  • Estes-Guralnick conjecture
  • Robinson's conjecture
  • Cyclotomic integers
  • Auxiliary function methods
  • Schur-Siegel-Smyth trace problem
  • Symmetrizable integer matrics
  • Signed and directed graphs
  • Salem numbers

Authors and Affiliations

  • Egham, UK

    James McKee

  • Edinburgh, UK

    Chris Smyth

About the authors

James McKee is Professor of Pure Mathematics at Royal Holloway, University of London. He is an expert on algorithmic and computational methods in number theory, particularly for elliptic curves, polynomials as well as Pisot and Salem numbers. In recent years his interests have become more combinatorial, and with his students and Smyth he has used structures related to graphs to study algebraic integers through their eigenvalues.

Chris Smyth, a professorial fellow in Number Theory at the University of Edinburgh, has a long-standing 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 2-dimensional Mahler measure involving an L-function, leading to a deep study of such formulae by Boyd, Deninger, Rodriguez Villegas and others. He invented the explicit auxiliary function method, which applies semi-infinite linear programming to number-theoretic 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/10.1007/978-3-030-80031-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2021

  • Softcover ISBN: 978-3-030-80030-7

  • eBook ISBN: 978-3-030-80031-4

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • 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

  • Industry Sectors: IT & Software

Buying options

eBook
USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-80031-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 79.99
Price excludes VAT (USA)