| Description |
1 online resource (xx, 438 pages : illustrations (some color)) |
| Series |
Universitext |
|
Universitext.
|
| Contents |
Mahler Measures of Polynomials in One Variable -- Mahler Measures of Polynomials in Several Variables -- Dobrowolski’s Theorem -- The Schinzel–Zassenhaus Conjecture -- Roots of Unity and Cyclotomic Polynomials -- Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem -- Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem -- The Set of Cassels Heights -- Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tessellations -- The Transfinite Diameter and Conjugate Sets of Algebraic Integers -- Restricted Mahler Measure Results -- The Mahler Measure of Nonreciprocal Polynomials -- Minimal Noncyclotomic Integer Symmetric Matrices -- The Method of Explicit Auxiliary Functions -- The Method of Explicit Auxiliary Functions -- Small-Span Integer Symmetric Matrices -- Symmetrizable Matrices I: Introduction -- Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices -- Symmetrizable Matrices III: The Trace Problem -- Salem Numbers from Graphs and Interlacing Quotients -- Minimal Polynomials of Integer Symmetric Matrices -- Breaking symmetry |
| 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 accessible to well-prepared upper-level undergraduates |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from PDF title page (SpringerLink, viewed December 22, 2021) |
| Subject |
Polynomials -- Measurement
|
|
Polinomios
|
|
Teoría algebraica de números
|
|
Polinomis.
|
| Genre/Form |
Electronic books
|
|
Llibres electrònics.
|
| Form |
Electronic book
|
| Author |
Smyth, Chris, author.
|
| ISBN |
9783030800314 |
|
3030800318 |
|
