Around the Unit Circle

1. Front Matter

   Pages i-xx

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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. Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem

   • James McKee, Chris Smyth

   Pages 113-136

8. Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem

   • James McKee, Chris Smyth

   Pages 137-147

9. The Set of Cassels Heights

   • James McKee, Chris Smyth

   Pages 149-169

10. Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tessellations

    • James McKee, Chris Smyth

    Pages 171-186

11. The Transfinite Diameter and Conjugate Sets of Algebraic Integers

    • James McKee, Chris Smyth

    Pages 187-203

12. Restricted Mahler Measure Results

    • James McKee, Chris Smyth

    Pages 205-218

13. The Mahler Measure of Nonreciprocal Polynomials

    • James McKee, Chris Smyth

    Pages 219-236

14. Minimal Noncyclotomic Integer Symmetric Matrices

    • James McKee, Chris Smyth

    Pages 237-248

15. The Method of Explicit Auxiliary Functions

    • James McKee, Chris Smyth

    Pages 249-268

16. The Trace Problem for Integer Symmetric Matrices

    • James McKee, Chris Smyth

    Pages 269-276

17. Small-Span Integer Symmetric Matrices

    • James McKee, Chris Smyth

    Pages 277-300

18. Symmetrizable Matrices I: Introduction

    • James McKee, Chris Smyth

    Pages 301-315

19. Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices

    • James McKee, Chris Smyth

    Pages 317-331

20. Symmetrizable Matrices III: The Trace Problem

    • James McKee, Chris Smyth

    Pages 333-342

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.

• 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

“It contains some material that is unavailable elsewhere. Each chapter is concluded by notes and a glossary of newly introduced definitions. … The reader at the graduate level having enough time and energy from this book can learn a lot about the Mahler measure, conjugate sets of algebraic integers and related results.” (Artūras Dubickas, zbMATH 1486.11003, 2022)

• Egham, UK

  James McKee

• Edinburgh, UK

  Chris Smyth

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.