Around the Unit Circle: Mahler Measure, Integer Matrices and Roots of Unity, Paperback

1 Mahler Measures of Polynomials in One Variable.- 2 Mahler Measures of Polynomials in Several Variables.- 3 Dobrowolski's Theorem.- 4 The Schinzel-Zassenhaus Conjecture.- 5 Roots of Unity and Cyclotomic Polynomials.- 6 Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem.- 7 Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem.- 8 The Set of Cassels Heights.- 9 Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tesselations.- 10 The Transfinite Diameter and Conjugate Sets of Algebraic Integers.- 11 Restricted Mahler Measure Results.- 12 The Mahler Measure of Nonreciprocal Polynomials.- 13 Minimal Noncyclotomic Integer Symmetric Matrices.- 14 The Method of Explicit Auxiliary Functions.- 15 The Trace Problem For Integer Symmetric Matrices.- 16 Small-Span Integer Symmetric Matrices.- 17 Symmetrizable Matrices I: Introduction.- 18 Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices.- 19 Symmetrizable Matrices III: The Trace Problem.- 20 Salem Numbers from Graphs and Interlacing Quotients.- 21 Minimal Polynomials of Integer Symmetric Matrices.- 22 Breaking Symmetry.- A Algebraic Background.- B Combinatorial Background.- C Tools from the Theory of Functions.- D Tables.- References.- Index.

About author(s):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.