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

The book discusses the Mahler measure, a height function for polynomials, and its properties, obtained through algebraic, analytic, and combinatorial methods. It covers various results on Mahler measure, including recent proofs and discussions of Robinson's Conjectures. The book is intended for lecture courses at the graduate level and will also be a valuable reference for researchers interested in Mahler measure. It is written in a friendly and accessible style, with numerous exercises and open problems.

Format: Paperback / softback
Length: 438 pages
Publication date: 09 December 2021
Publisher: Springer Nature Switzerland AG

The Mahler measure, a height function for polynomials, forms the central focus of this book. It possesses numerous intriguing properties, which have been derived through algebraic, analytic, and combinatorial methodologies. Numerous longstanding unsolved questions, such as Lehmers Problem (1933) and Boyds Conjecture (1981), revolve around it. This comprehensive volume presents a diverse array of results on Mahler measure. Some of these findings are quite recent, exemplified by Dimitrov's proof of the Schinzel–Zassenhaus Conjecture. Additionally, well-known results are accompanied by streamlined proofs. The book also delves into Robinson's Conjectures (1965) for cyclotomic integers and their associated Cassels height function, a topic that has hitherto been overlooked in literature.

One approach to studying algebraic integers involves associating them with combinatorial objects, such as integer matrices. In certain combinatorial settings, analogues of several notorious open problems have been resolved, and the book outlines this recent work. Numerous Mahler measure results are established for restricted sets of polynomials, including totally real polynomials and reciprocal polynomials of integer symmetric and symmetrizable matrices. For reference, the book includes appendices that provide essential 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 thoroughly motivated and presented in a user-friendly manner. Numerous exercises at various levels are provided, including some for computer programming. A wide range of stimulating open problems is also included. At the conclusion of each chapter, a glossary of newly introduced concepts and definitions is provided.

Around the Unit Circle is authored in a friendly, lucid, and enjoyable style, while maintaining rigorous mathematical precision. Its primary objective is to captivate and engage readers, particularly those with an interest in mathematics.

Weight: 698g
Dimension: 157 x 234 x 30 (mm)
ISBN-13: 9783030800307
Edition number: 1st ed. 2021