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001441272 020__ $$a9783030800314$$q(electronic bk.)
001441272 020__ $$a3030800318$$q(electronic bk.)
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001441272 0247_ $$a10.1007/978-3-030-80031-4$$2doi
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001441272 050_4 $$aQA161.P59$$bM35 2021eb
001441272 08204 $$a512.9/422$$223
001441272 1001_ $$aMcKee, James$$q(James Fraser),$$eauthor.
001441272 24510 $$aAround the unit circle :$$bMahler measure, integer matrices and roots of unity /$$cJames McKee, Chris Smyth.
001441272 264_1 $$aCham :$$bSpringer,$$c[2021]
001441272 264_4 $$c©2021
001441272 300__ $$a1 online resource (xx, 438 pages : illustrations (some color))
001441272 336__ $$atext$$btxt$$2rdacontent
001441272 337__ $$acomputer$$bc$$2rdamedia
001441272 338__ $$aonline resource$$bcr$$2rdacarrier
001441272 4901_ $$aUniversitext
001441272 504__ $$aIncludes bibliographical references and index.
001441272 50500 $$tMahler Measures of Polynomials in One Variable --$$tMahler Measures of Polynomials in Several Variables --$$tDobrowolski's Theorem --$$tThe Schinzel-Zassenhaus Conjecture --$$tRoots of Unity and Cyclotomic Polynomials --$$tCyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem --$$tCyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem --$$tThe Set of Cassels Heights --$$tCyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tessellations --$$tThe Transfinite Diameter and Conjugate Sets of Algebraic Integers --$$tRestricted Mahler Measure Results --$$tThe Mahler Measure of Nonreciprocal Polynomials --$$tMinimal Noncyclotomic Integer Symmetric Matrices --$$tThe Method of Explicit Auxiliary Functions --$$tThe Method of Explicit Auxiliary Functions --$$tSmall-Span Integer Symmetric Matrices --$$tSymmetrizable Matrices I: Introduction --$$tSymmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices --$$tSymmetrizable Matrices III: The Trace Problem --$$tSalem Numbers from Graphs and Interlacing Quotients --$$tMinimal Polynomials of Integer Symmetric Matrices --$$tBreaking symmetry.
001441272 506__ $$aAccess limited to authorized users.
001441272 520__ $$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.
001441272 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed December 22, 2021).
001441272 650_0 $$aPolynomials$$xMeasurement.
001441272 655_0 $$aElectronic books.
001441272 7001_ $$aSmyth, Chris,$$eauthor.
001441272 77608 $$iPrint version:$$aMcKee, James (James Fraser).$$tAround the unit circle.$$dCham : Springer, [2021]$$z303080030X$$z9783030800307$$w(OCoLC)1253474622
001441272 830_0 $$aUniversitext.
001441272 852__ $$bebk
001441272 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-030-80031-4$$zOnline Access$$91397441.1
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001441272 980__ $$aEBOOK
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