000			a
999			_c30722 _d30722
008			220222b xxu||||| |||| 00| 0 eng d
020			_a9783030800307
082			_a512.9422 _bMCK
100			_aMcKee, James
245			_aAround the unit circle : Mahler measure, integer matrices and roots of unity
260			_bSpringer, _c2021 _aCham :
300			_axx, 438 p. ; _bill., _c24 cm
365			_b64.99 _cEUR _d88.10
490			_aUniversitext
504			_aIncludes bibliographical references and index.
520			_aMahler 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.
650			_aNumber Theory
650			_aLinear Algebra
650			_aPolynomials
650			_aMeasurement
650			_aGraph Theory
650			_aBogomolov constant
650			_a Cassels
650			_a Common neighbour class
650			_aConjugate set
650			_aInteger symmetric matrix
650			_aDimitrov's Theorem
650			_a Mahler measure
650			_a Cyclotomic integers
650			_a Estes-Guralick Conjecture
650			_a Fermat's Little Theorem
650			_aHardy function
650			_a Interlacing Theorem
650			_aKronecker's Theorem
650			_a Lehmer's Conjecture
650			_aMonic polynomial
650			_aPisot number
650			_aRouche's Theorem
650			_aToroidal tessellation
700			_aSmyth, Chris
942			_2ddc _cBK