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Author: François Brunault
Publisher: Cambridge University Press
ISBN: 1108889190
Category : Mathematics
Languages : en
Pages : 185
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Book Description
The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne–Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.
Author: François Brunault
Publisher: Cambridge University Press
ISBN: 1108889190
Category : Mathematics
Languages : en
Pages : 185
Get Book Here
Book Description
The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne–Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.
Author: James McKee
Publisher: Springer Nature
ISBN: 3030800318
Category : Mathematics
Languages : en
Pages : 444
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Book Description
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.
Author: Basil Gordon
Publisher: Springer Science & Business Media
ISBN: 9780792355830
Category : Computers
Languages : en
Pages : 280
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Book Description
This volume contains the proceedings of the Topics in Number Theory Conference held at the Pennsylvania State University from July 31 through August 3, 1997. It contains seventeen research papers covering many areas of number theory; among them are contributions from four of the eight plenary speakers
Author: Graham Everest
Publisher: Springer Science & Business Media
ISBN: 1447138988
Category : Mathematics
Languages : en
Pages : 217
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Book Description
The main theme of this book is the theory of heights as they appear in various guises. This includes a large body of results on Mahlers measure of the height of a polynomial. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations, with all special calculation included so as to be self-contained. The authors devote space to discussing Mahlers measure and to giving some convincing and original examples to explain this phenomenon. XXXXXXX NEUER TEXT The main theme of this book is the theory of heights as it appears in various guises. To this §End.txt.Int.:, it examines the results of Mahlers measure of the height of a polynomial, which have never before appeared in book form. The authors take a down-to-earth approach that includes convincing and original examples. The book uncovers new and interesting connections between number theory and dynamics and will be interesting to researchers in both number theory and nonlinear dynamics.
Author: A. Schinzel
Publisher: Cambridge University Press
ISBN: 9781139426718
Category : Mathematics
Languages : en
Pages : 590
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Book Description
This book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 64
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Book Description
Author: Thomas Peattie
Publisher: Cambridge University Press
ISBN: 1316298442
Category : Music
Languages : en
Pages : 233
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Book Description
In this study Thomas Peattie offers a new account of Mahler's symphonies by considering the composer's reinvention of the genre in light of his career as a conductor and more broadly in terms of his sustained engagement with the musical, theatrical, and aesthetic traditions of the Austrian fin de siècle. Drawing on the ideas of landscape, mobility, and theatricality, Peattie creates a richly interdisciplinary framework that reveals the uniqueness of Mahler's symphonic idiom and its radical attitude toward the presentation and ordering of musical events. The book goes on to identify a fundamental tension between the music's episodic nature and its often-noted narrative impulse and suggests that Mahler's symphonic dramaturgy can be understood as a form of abstract theatre.
Author: James Fraser McKee
Publisher: Cambridge University Press
ISBN: 0521714672
Category : Mathematics
Languages : en
Pages : 350
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Book Description
Contributions by leading experts in the field provide a snapshot of current progress in polynomials and number theory.
Author: Wadim Zudilin
Publisher: World Scientific
ISBN: 9811279330
Category : Mathematics
Languages : en
Pages : 192
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Book Description
There is no surprise that arithmetic properties of integral ('whole') numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other 'better known' numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of π and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible — almost popular — level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.
Author: Brid ni Fhlathuin
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
