Around the unit circle : Mahler measure, integer matrices and roots of unity / James McKee, Chris Smyth.

Format:

Book

Author/Creator:

McKee, James (James Fraser), author.

Smyth, Chris, author.

Series:

Universitext

Language:

English

Subjects (All):

Polynomials.

Measurement.

Polynomials--Measurement.

Genre:

Electronic books.

Physical Description:

1 online resource : illustrations (some color).

Place of Publication:

Cham : Springer, [2021]

System Details:

text file

Summary:

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 Lehmers Problem (1933) and Boyds Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrovs proof of the SchinzelZassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinsons 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 accessibl e to well-prepared upper-level undergraduates.

Contents:

Intro

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

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

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

Notes:

Includes bibliographical references and index.

Online resource; title from PDF title page (SpringerLink, viewed December 22, 2021).

Other Format:

Print version: McKee, James (James Fraser). Around the unit circle.

ISBN:

9783030800314

3030800318

OCLC:

1288168246