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everest graham; ward thomas - heights of polynomials and entropy in algebraic dynamics

Heights of Polynomials and Entropy in Algebraic Dynamics

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Dettagli

Genere:Libro
Lingua: Inglese
Editore:

Springer

Pubblicazione: 10/2010
Edizione: Softcover reprint of hardcover 1st ed. 1999





Trama

Arithmetic geometry and algebraic dynamical systems are flourishing areas of mathematics. Both subjects have highly technical aspects, yet both of­ fer a rich supply of down-to-earth examples. Both have much to gain from each other in techniques and, more importantly, as a means for posing (and sometimes solving) outstanding problems. It is unlikely that new graduate students will have the time or the energy to master both. This book is in­ tended as a starting point for either topic, but is in content no more than an invitation. We hope to show that a rich common vein of ideas permeates both areas, and hope that further exploration of this commonality will result. Central to both topics is a notion of complexity. In arithmetic geome­ try 'height' measures arithmetical complexity of points on varieties, while in dynamical systems 'entropy' measures the orbit complexity of maps. The con­ nections between these two notions in explicit examples lie at the heart of the book. The fundamental objects which appear in both settings are polynomi­ als, so we are concerned principally with heights of polynomials. By working with polynomials rather than algebraic numbers we avoid local heights and p-adic valuations.




Sommario

1. Lehmer, Mahler and Jensen.- 2. Dynamical Systems.- 3. Mahler’s Measure in Many Variables.- 4. Higher-Dimensional Dynamical Systems.- 5. Elliptic Heights.- 6. The Elliptic Mahler Measure.- A. Algebra.- A.1 Algebraic Integers.- A.2 Integer Matrices.- A.3 Hilbert’s Nullstellensatz.- B. Analysis.- B.1 Stone-Weierstrass Theorem.- B.2 The Gelfand Transform.- C. Division Polynomials.- E.1 Lehmer Primes.- E.2 Elliptic Primes.- F. Exercises and Questions.- F.1 Hints for the Exercises.- F.2 List of Questions.- G. List of Notation.










Altre Informazioni

ISBN:

9781849968546

Condizione: Nuovo
Collana: Universitext
Dimensioni: 279 x 210 mm Ø 571 gr
Formato: Brossura
Illustration Notes:XII, 212 p.
Pagine Arabe: 212
Pagine Romane: xii


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