Heights of Polynomials and Entropy in Algebraic Dynamics

by Everest, Graham; Ward, Thomas

• ISBN13: 9781852331252
• ISBN10: 1852331259
• Format: Hardcover
• Copyright: 1999-04-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Lehmer, Mahler and Jensen	1	(28)
Background: Lehmer's Paper	1	(2)
Mahler's Measure	3	(10)
Lehmer's Problem I: Schinzel's Theorem	13	(2)
Lehmer's Problem II: Smyth's Theorem	15	(6)
Lehmer's Problem III: Zhang's Theorem	21	(4)
Large Primes in 1933	25	(2)
When Does the Measure Vanish?	27	(2)
Dynamical Systems	29	(22)
Dynamical Interpretation: Toral Case	29	(5)
Topological Entropy	34	(5)
Dynamical Interpretation: Solenoid Case	39	(9)
Periodic Points in the Dynamical Interpretation	48	(3)
Mahler's Measure in Many Variables	51	(30)
Explicit Values	52	(5)
Existence	57	(2)
When Does the Measure Vanish?	59	(10)
Approximations to Two-Dimensional Mahler Measure	69	(4)
Lawton's Estimate	73	(6)
Boyd's Example	79	(2)
Higher-Dimensional Dynamical Systems	81	(10)
The Dynamical System Associated to a Polynomial	81	(1)
Mahler Measures as Entropies	82	(4)
Periodic Point Behaviour	86	(3)
Dynamical Zeta Functions	89	(2)
Elliptic Heights	91	(26)
Elliptic Curves	91	(4)
Explicit Formulæ	95	(4)
Torsion Points of Order 11	99	(1)
Rational Points	100	(1)
Heights on Elliptic Curves	101	(5)
Mordell's Theorem	106	(3)
The Parallelogram Law and the Canonical Height	109	(5)
Heights of Algebraic Numbers	114	(3)
The Elliptic Mahler Measure	117	(28)
Elliptic Functions	117	(7)
Elliptic Mahler	124	(7)
Elliptic Mahler in Several Variables	131	(6)
Elliptic Mahler and Periodic Points	137	(3)
Application to the Division Polynomial	140	(2)
Elliptic Mahler and Dynamical Systems	142	(3)
A. Algebra	145	(6)
A.1 Algebraic Integers	145	(2)
A.2 Integer Matrices	147	(1)
A.3 Hilbert's Nullstellensatz	148	(3)
B. Analysis	151	(6)
B.1 Stone--Weierstrass Theorem	151	(3)
B.2 The Gelfand Transform	154	(3)
C. Division Polynomials	157	(2)
D. Proof of Mahler's Bound for m(F')	159	(6)
E. Calculations on Δn (F) and En (F)	165	(4)
E.1 Lehmer Primes	165	(1)
E.2 Elliptic Primes	166	(3)
F. Exercises and Questions	169	(28)
F.1 Hints for the Exercises	169	(25)
F.2 List of Questions	194	(3)
G. List of Notation	197	(2)
Bibliography	199	(10)
Index	209