Overview

The main theme of the book is the theory of heights as they appear in various guises. This includes a large body of results on Mahler's measure of the height of a polynomial of which topic there is no book available. The genesis of the measure in a paper by Lehmer is looked at, which is extremely well-timed due to the revival of interest following the work of Boyd and Deninger on special values of Mahler's measure. 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. A large chunk of the book has been devoted to the elliptic Mahler's measure. Special calculation have been included and will be self-contained. One of the most important results about Mahler's measure is that it is the entropy associated to a dynamical system. The authors devote space to discussing this and to giving some convincing and original examples to explain this phenomenon.

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9781852331252

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From the reviews: At first sight it would seem exceedingly unlikely that there would be any relation at all between entropy and height. Reality is otherwise: they are very much intertwined, and this is what this unusual and very interesting book is about. ... The authors write about all this with erudition and charm ... . They have included more than a hundred exercises ... . (M. Hazewinkel, Nieuw Archief voor Wiskunde, Vol. 5/6 (3), 2005) It ranges through a number of topics, varying from the elementary to the sophisticated, all featuring polynomials. ... this text provides an excellent basis for a beginning postgraduate course, since most of the material is not too demanding and yet it arouses the curiosity to learn more about dynamical systems, algebraic number theory or primality testing. In addition, there are 103 exercises with hints, five useful appendices sketching the prerequisites, and an extensive bibliography ... . (Victor P. Snaith, Bulletin of the London Mathematical Society, Vol. 32, 2000) This book provides an introduction to algebraic dynamical systems and their connection with number theory. This relationship is illustrated by many examples ... . I recommend the reading of this interesting book, which may be accessible even for advanced undergraduate students. (Yann Bugeaud, Mathematical Reviews, Issue 2000 e) This unusual book is based on a course given to postgraduate students at the University of East Anglia. It could serve as a textbook or as interesting supplementary reading for a course in algebraic dynamics or elliptic curves. ... the authors follow the sound pedagogical practice of giving detailed proofs of special cases and providing references to the literature for more general results. (D. W. Boyd, Zentralblatt MATH, Vol. 919, 1999) This monograph explores two notions of height in the context of dynamics: Mahler measure and elliptic height. ... the material of the book is very well presented and the historical references, examples and exercises make it a useful text for anybody trying to get into the spirit of algebraic dynamical systems. (K. Schmidt, Monatshefte fur Mathematik, Issue 3, 1999)

From the reviews: <p> At first sight it would seem exceedingly unlikely that there would be any relation at all between entropy and height. Reality is otherwise: they are very much intertwined, and this is what this unusual and very interesting book is about. a ] The authors write about all this with erudition and charm a ] . They have included more than a hundred exercises a ] . (M. Hazewinkel, Nieuw Archief voor Wiskunde, Vol. 5/6 (3), 2005) <p> It ranges through a number of topics, varying from the elementary to the sophisticated, all featuring polynomials. a ] this text provides an excellent basis for a beginning postgraduate course, since most of the material is not too demanding and yet it arouses the curiosity to learn more about dynamical systems, algebraic number theory or primality testing. In addition, there are 103 exercises with hints, five useful appendices sketching the prerequisites, and an extensive bibliography a ] . (Victor P. Snaith, Bulletin of the London Mathematical Society, Vol. 32, 2000) <p> This book provides an introduction to algebraic dynamical systems and their connection with number theory. This relationship is illustrated by many examples a ] . I recommend the reading of this interesting book, which may be accessible even for advanced undergraduate students. (Yann Bugeaud, Mathematical Reviews, Issue 2000 e) <p> This unusual book is based on a course given to postgraduate students at the University of East Anglia. It could serve as a textbook or as interesting supplementary reading for a course in algebraic dynamics or elliptic curves. a ] the authors follow the sound pedagogical practice of giving detailed proofs of special cases and providingreferences to the literature for more general results. (D. W. Boyd, Zentralblatt MATH, Vol. 919, 1999) <p> This monograph explores two notions of height in the context of dynamics: Mahler measure and elliptic height. a ] the material of the book is very well presented and the historical references, examples and exercises make it a useful text for anybody trying to get into the spirit of algebraic dynamical systems. (K. Schmidt, Monatshefte fA1/4r Mathematik, Issue 3, 1999)

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