4/17/2023 0 Comments Iunit circleHe is an expert on algorithmic and computational methods in number theory, particularly for elliptic curves, polynomials as well as Pisot and Salem numbers. James McKee is Professor of Pure Mathematics at Royal Holloway, University of London. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. At the end of each chapter there is a glossary of newly introduced concepts and definitions.Īround the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. A wide range of stimulating open problems is also included. Numerous exercises at various levels are given, including some for computer programming. All theorems are well motivated and presented in an accessible way. 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. 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. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book. Other known results are included with new, streamlined proofs. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. This book contains a wide range of results on Mahler measure. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. Mahler measure, a height function for polynomials, is the central theme of this book.
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