Around the unit circle [Documento eletrónico] : Mahler measure, integer matrices and roots of unity / by James McKee, Chris Smyth
Language: eng.Country: Switzerland, Swiss Confederation.Publication: Cham : Springer, 2021Description: XX, 438 p. : il.ISBN: 978-3-030-80031-4.Series: UniversitextSubject - Topical Name: Number theory | Graph theory | Algebras, Linear Online Resources:Click here to access onlineItem type | Current library | Collection | Call number | Copy number | Status | Date due | Barcode | |
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QA241.SPR FCT Cubic forms and the circle method | QA241.SPR FCT Excursions in number theory, algebra, and analysis | QA241.SPR FCT Advances in non-archimedean analysis and applications, the p-adic methodology in STEAM-H | QA241.SPR FCT Around the unit circle, Mahler measure, integer matrices and roots of unity | QA241.SPR FCT Number theory in memory of Eduard Wirsing | QA241.SPR FCT Women in numbers europe III, research directions in number theory | QA241.SPR FCT A textbook of algebraic number theory |
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 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.
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