---

＜電子ブック＞
Around the Unit Circle : Mahler Measure, Integer Matrices and Roots of Unity / by James McKee, Chris Smyth
(Universitext. ISSN:21916675)

版	1st ed. 2021.
出版者	(Cham : Springer International Publishing : Imprint: Springer)
出版年	2021
本文言語	英語
大きさ	XX, 438 p. 98 illus., 2 illus. in color : online resource
著者標目	*McKee, James author Smyth, Chris author SpringerLink (Online service)
件　名	LCSH:Number theory LCSH:Graph theory LCSH:Algebras, Linear FREE:Number Theory FREE:Graph Theory FREE:Linear Algebra
一般注記	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 -- 19Symmetrizable 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 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 setsof 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 HTTP:URL=https://doi.org/10.1007/978-3-030-80031-4

 類似資料