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

Preface
Contents
1 Mahler Measures of Polynomials in One Variable
1.1 Introduction
1.1.1 Polynomials over the Field mathbbC of Complex Numbers
1.1.2 Polynomials over the Field mathbbQ of Rational Numbers
1.2 Kronecker's Two Theorems
1.3 Mahler Measure Inequalities
1.4 A Lower Bound for an Integer Polynomial Evaluated at an Algebraic Number
1.5 Polynomials with Small Coefficients
1.6 Separation of Conjugates
1.7 The Shortness of a Polynomial
1.7.1 Finding Short Polynomials
1.8 Variants of Mahler Measure
1.8.1 The Weil Height
1.9 Notes
1.10 Glossary
2 Mahler Measures of Polynomials in Several Variables
2.1 Introduction
2.2 Preliminaries for the Proofs of Theorems 2.5 and 2.6
2.3 Proof of Theorem 2.5
2.4 Proof of Theorem 2.6
2.5 Computation of Two-Dimensional Mahler Measures
2.6 Small Limit Points of mathcalL?
2.6.1 Shortness Conjectures Implying Lehmer's Conjecture and Structural Results for mathcalL
2.6.2 Small Elements of the Set of Two-Variable Mahler Measures
2.7 Closed Forms for Mahler Measures of Polynomials of Dimension at Least 2
2.7.1 Dirichlet L-Functions
2.7.2 Some Explicit Formulae for Two-Dimensional Mahler Measures
2.7.3 Mahler Measures of Elliptic Curves
2.7.4 Mahler Measure of Three-Dimensional Polynomials
2.7.5 Mahler Measure Formulae for Some Polynomials of Dimension at Least 4
2.7.6 An Asymptotic Mahler Measure Result
2.8 Notes
2.9 Glossary
3 Dobrowolski's Theorem
3.1 The Theorem and Preliminary Lemmas
3.2 Proof of Theorem 3.1: Dobrowolski's Lower Bound for M(α)
3.3 Notes
3.4 Glossary
4 The Schinzel–Zassenhaus Conjecture
4.1 Introduction
4.1.1 A Simple Proof of a Weaker Result
4.2 Proof of Dimitrov's Theorem
4.3 Notes
4.4 Glossary
5 Roots of Unity and Cyclotomic Polynomials
5.1 Introduction
5.2 Solving Polynomial Equations in Roots of Unity
5.3 Cyclotomic Points on Curves
5.3.1 Definitions
5.3.2 mathcalL(f) of Rank 1
5.3.3 mathcalL(f) Full of Rank 2
5.3.4 mathcalL(f) of Rank 2, but Not Full
5.3.5 The Case of f Reducible
5.3.6 An Example
5.4 Cyclotomic Integers
5.4.1 Introduction to Cyclotomic Integers
5.4.2 The Function mathscrN(β)
5.4.3 Evaluating or Estimating mathscrN(sqrtd)
5.4.4 Evaluation of the Gauss Sum
5.4.5 The Absolute Mahler Measure of Cyclotomic Integers
5.5 Robinson's Problems and Conjectures
5.6 Cassels' Lemmas for mathscrM(β)
5.7 Discussion of Robinson's Problems
5.7.1 Robinson's First Problem
5.7.2 Robinson's Second Problem
5.8 Discussion of Robinson's Conjectures
5.8.1 The First Conjecture
5.8.2 The Second Conjecture
5.8.3 The Third Conjecture
5.8.4 The Fourth Conjecture
5.8.5 The Fifth Conjecture
5.9 Multiplicative Relations Between Conjugate Roots of Unity
5.10 Notes
5.11 Glossary
6 Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem
6.1 Introduction
6.2 The Mahler Measure of a Matrix and Cyclotomic Matrices
6.3 Flavours of Equivalence: Isomorphism, Equivalence and Strong Equivalence of Matrices
6.4 Growing Cyclotomic Matrices
6.5 Gram Vectors
6.6 Statement of the Classification Theorem for Cyclotomic Integer Symmetric Matrices
6.7 Glossary
7 Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem
7.1 Cyclotomic Signed Graphs
7.2 Cyclotomic Charged Signed Graphs
7.3 Cyclotomic Integer Symmetric Matrices: Completion of the Classification
7.4 Further Exercises
7.5 Notes on Chaps. 6摥映數爠eflinkC:CYCLOTOMICS66 and 7
7.6 Glossary
8 The Set of Cassels Heights
8.1 Cassels Height and the Set mathscrC
8.2 The Derived Sets and the Sumsets of mathscrC
8.3 Proof of Theorem 8.4
8.3.1 Structure and Labelling of Thue Sets
8.4 Cassels Heights of Cyclotomic Integers in mathbbQ(ωp)
8.5 Proof of Theorem 8.14
8.6 Proof of Theorem 8.13
8.7 Notes
8.8 Glossary
9 Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tessellations
9.1 Introduction
9.2 Preliminaries: Notation and Tools
9.3 Cyclotomic Graphs Embedded in T2k
9.4 Changes for Charges
9.5 Glossary
10 The Transfinite Diameter and Conjugate Sets of Algebraic Integers
10.1 Introduction
10.2 Analytic Properties of the Transfinite Diameter
10.3 Application to Conjugate Sets of Algebraic Integers
10.4 Integer Transfinite Diameters
10.4.1 The Integer Transfinite Diameter
10.4.2 The Monic Integer Transfinite Diameter
10.5 Notes
10.6 Glossary
11 Restricted Mahler Measure Results
11.1 Monic Integer Irreducible Noncyclotomic Polynomials
11.2 Complex Polynomials That are Sums of a Bounded Number of Monomials
11.3 Some Sets of Algebraic Numbers with the Bogomolov Property
11.3.1 Totally p-Adic Fields
11.3.2 Abelian Extensions of mathbbQ
11.3.3 Langevin's Theorem
11.4 The Height of Zhang and Zagier and Generalisations
11.5 The Weil Height of α When mathbbQ(α)/mathbbQ is Galois
11.6 Notes
11.7 Glossary
12 The Mahler Measure of Nonreciprocal Polynomials
12.1 Mahler Measure of Nonreciprocal Polynomials
12.1.1 The Set mathcalH of Rational Hardy Functions
12.2 Proof of Theorem 12.1
12.2.1 Start of the Proof
12.2.2 The Case ell< 2k
12.2.3 The Case ellge2k: Proof that M(P)geθ0
12.2.4 The Case ellge2k: Existence of a δ, Part 1
12.2.5 The Case ellge2k: Existence of a δ, Part 2
12.2.6 The Case ellge2k: Completion of the Proof
12.3 Notes
12.4 Glossary
13 Minimal Noncyclotomic Integer Symmetric Matrices
13.1 Supersporadic Matrices and Other Sporadic Examples
13.2 Minimal Noncyclotomic Charged Signed Graphs: Any that Are Not Supersporadic
13.2.1 The Uncharged Case
13.2.2 The Charged Case
13.3 Completing the Classification
13.4 Notes
13.5 Glossary
14 The Method of Explicit Auxiliary Functions
14.1 Conjugate Sets of Algebraic Numbers
14.2 The Optimisation Problem
14.2.1 Dualising the Problem
14.2.2 Method Outline
14.3 The Schur–Siegel–Smyth Trace Problem
14.3.1 Totally Positive Algebraic Integers with Small Mean Trace
14.4 The Mean Trace of α Less Its Least Conjugate
14.5 An Upper Bound Trace Problem
14.6 Mahler Measure of Totally Real Algebraic Integers
14.7 Mahler Measure of Totally Real Algebraic Numbers
14.8 Langevin's Theorem for Sectors
14.8.1 Further Remarks
14.9 Notes
14.10 Glossary
15 The Trace Problem for Integer Symmetric Matrices
15.1 The Mean Trace of a Positive Definite Matrix
15.2 The Trace Problem for Integer Symmetric Matrices
15.3 Constructing Examples that Have Minimal Trace
15.4 Notes
15.5 Glossary
16 Small-Span Integer Symmetric Matrices
16.1 Small-Span Polynomials
16.2 Small-Span Integer Symmetric Matrices
16.3 Bounds on Entries and Degrees
16.4 Growing Small Examples
16.4.1 Two Rows
16.4.2 Three Rows
16.4.3 Four Rows
16.4.4 Five Rows
16.4.5 Six Rows
16.4.6 Seven Rows
16.4.7 Eight Rows
16.4.8 Nine Rows
16.4.9 Ten Rows
16.4.10 Eleven Rows
16.4.11 Twelve Rows
16.4.12 Thirteen Rows
16.5 Cyclotomic Small-Span Matrices
16.5.1 Examples with an Entry of Modulus Greater Than 1
16.5.2 Subgraphs of the Sporadic Examples
16.5.3 Subgraphs of Cylindrical Tessellations
16.5.4 Subgraphs of Toroidal Tessellations
16.6 The Classification Theorem
16.7 Notes
16.8 Glossary
17 Symmetrizable Matrices I: Introduction
17.1 Introduction
17.2 Definitions and Immediate Consequences
17.3 The Structure of Symmetrizable Matrices
17.4 The Balancing Condition and Its Consequences
17.5 The Symmetrization Map
17.6 Interlacing
17.7 Equitable Partitions of Signed Graphs
17.8 Notes
17.9 Glossary
18 Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices
18.1 Cyclotomic Symmetrizable Integer Matrices
18.2 Quotients of Signed Graphs
18.3 Notes
18.4 Glossary
19 Symmetrizable Matrices III: The Trace Problem
19.1 The Trace Problem for Symmetrizable Matrices
19.1.1 Definitions, Notation and Statement of the Results
19.1.2 Proof of Proposition 19.3摥映數爠eflinkP:rattrace19.319
19.1.3 Corollaries, Including Theorem 19.4摥映數爠eflinkT:main19.419
19.1.4 The Structure of Minimal-Trace Examples
19.2 Notes
19.3 Glossary
20 Salem Numbers from Graphs and Interlacing Quotients
20.1 Introduction
20.2 Salem Graphs
20.3 Examples of Salem Graphs
20.3.1 Nonbipartite Examples
20.3.2 Bipartite Examples
20.3.3 Finding Cyclotomic Factors
20.4 Attaching Pendant Paths
20.4.1 A General Construction
20.4.2 An Application to Salem Graphs
20.5 Interlacing Quotients
20.5.1 Rational Interlacing Quotients
20.5.2 Circular Interlacing Quotients
20.5.3 From CIQs to Cyclotomic RIQs
20.5.4 Salem Numbers from Interlacing Quotients
20.6 Notes
20.7 Glossary
21 Minimal Polynomials of Integer Symmetric Matrices
21.1 Introduction
21.2 Small Discriminant
21.3 Small Span
21.3.1 The Cyclotomic Case
21.3.2 The Noncyclotomic Case
21.3.3 Some Counterexamples to Conjecture21.2
21.4 Small Trace
21.5 Polynomials that are Not Interlaced
21.6 Counterexamples for all Degrees Greater than 5
21.6.1 Degrees 8 to 16
21.6.2 Degree 20
21.6.3 Degree 19 and All Degrees Greater than 20
21.6.4 All Together Now
21.7 Notes
21.8 Glossary
22 Breaking Symmetry
Appendix A Algebraic Background
A.1 Self-Reciprocal Polynomials
A.2 Resultant Essentials
A.3 Valuation Essentials
A.4 Galois Theory Essentials
A.5 Algebraic Numbers and Algebraic Integers
A.5.1 The Gorškov Polynomials
A.6 Newton's Identities
A.7 Jänichen's Generalisation of Fermat's Little Theorem
Appendix B Combinatorial Background
B.1 Interlacing
B.2 Graph Theory
B.3 Perron–Frobenius Theory
Appendix C Tools from the Theory of Functions
Appendix D Tables
D.1 Small Mahler Measures
D.2 Known Small Mahler Measures of Two-Variable Polynomials
Appendix References
Index