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Dynamical Mordell-Lang Conjecture [Kõva köide]

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  • Formaat: Hardback, 280 pages, kõrgus x laius: 254x178 mm, kaal: 674 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470424088
  • ISBN-13: 9781470424084
Teised raamatud teemal:
Dynamical Mordell-Lang ConjectureSuurem pilt
  • Formaat: Hardback, 280 pages, kõrgus x laius: 254x178 mm, kaal: 674 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470424088
  • ISBN-13: 9781470424084
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470424084.html
Märksõnad:
In this book, authors Bell, Ghioca, and Tucker present readers with an in-depth look at the dynamical Mordell-Lang conjecture, presenting all of its known results. The fifteen chapters that make up the main bulk of the text are focused on the back ground mathematical materials for understanding the conjecture, the dynamical Mordell-Lang problem, a geometric Skolem-Mahler-Lech theorem, linear relations between points in polynomial orbits, and a variety of other related subjects. Jason P. Bell is a faculty member of the University of Waterloo in Canada. Dragos Ghioca is a faculty member of the University of British Columbia in Canada. Thomas J. Tucker is a faculty member of the University of Rochester, New York. Annotation ©2016 Ringgold, Inc., Portland, OR (protoview.com)
Preface xi
Notation xii
Chapter 1 Introduction
1(10)
1.1 Overview of the problem
1(1)
1.2 Linear recurrence sequences
2(1)
1.3 Polynomial-exponential Diophantine equations
3(1)
1.4 Linear algebra
4(1)
1.5 Arithmetic geometry
5(2)
1.6 Plan of the book
7(4)
Chapter 2 Background material
11(36)
2.1 Algebraic geometry
11(11)
2.2 Dynamics of endomorphisms
22(2)
2.3 Valuations
24(8)
2.4 Chebotarev Density Theorem
32(1)
2.5 The Skolem-Mahler-Lech Theorem
33(7)
2.6 Heights
40(7)
Chapter 3 The Dynamical Mordell-Lang problem
47(20)
3.1 The Dynamical Mordell-Lang Conjecture
47(5)
3.2 The case of rational self-maps
52(2)
3.3 Known cases of the Dynamical Mordell-Lang Conjecture
54(3)
3.4 The Mordell-Lang conjecture
57(4)
3.5 Denis-Mordell-Lang conjecture
61(1)
3.6 A more general Dynamical Mordell-Lang problem
62(5)
Chapter 4 A geometric Skolem-Mahler-Lech Theorem
67(18)
4.1 Geometric reformulation
67(1)
4.2 Automorphisms of affine varieties
68(2)
4.3 Etale maps
70(3)
4.4 Proof of the Dynamical Mordell-Lang Conjecture for etale maps
73(12)
Chapter 5 Linear relations between points in polynomial orbits
85(32)
5.1 The main results
85(4)
5.2 Intersections of polynomial orbits
89(2)
5.3 A special case
91(2)
5.4 Proof of Theorem 5.3.0.2
93(5)
5.5 The general case of Theorem 5.3.0.1
98(1)
5.6 The method of specialization and the proof of Theorem 5.5.0.2
98(4)
5.7 The case of Theorem 5.2.0.1 when the polynomials have different degrees
102(5)
5.8 An alternative proof for the function field case
107(3)
5.9 Possible extensions
110(1)
5.10 The case of plane curves
110(3)
5.11 A Dynamical Mordell-Lang type question for polarizable endomorphisms
113(4)
Chapter 6 Parametrization of orbits
117(10)
6.1 Rational maps
118(2)
6.2 Analytic uniformization
120(4)
6.3 Higher dimensional parametrizations
124(3)
Chapter 7 The split case in the Dynamical Mordell-Lang Conjecture
127(16)
7.1 The case of rational maps without periodic critical points
129(3)
7.2 Extension to polynomials with complex coefficients
132(4)
7.3 The case of "almost" post-critically finite rational maps
136(7)
Chapter 8 Heuristics for avoiding ramification
143(10)
8.1 A random model heuristic
143(3)
8.2 Random models and cycle lengths
146(2)
8.3 Random models and avoiding ramification
148(2)
8.4 The case of split maps
150(3)
Chapter 9 Higher dimensional results
153(14)
9.1 The Herman-Yoccoz method for periodic attracting points
153(5)
9.2 The Herman-Yoccoz method for periodic indifferent points
158(1)
9.3 The case of semiabelian varieties
159(1)
9.4 Preliminaries from linear algebra
160(2)
9.5 Proofs for Theorems 9.2.0.1 and 9.3.0.1
162(5)
Chapter 10 Additional results towards the Dynamical Mordell-Lang Conjecture
167(12)
10.1 A v-adic analytic instance of the Dynamical Mordell-Lang Conjecture
167(4)
10.2 A real analytic instance of the Dynamical Mordell-Lang Conjecture
171(4)
10.3 Birational polynomial self-maps on the affine plane
175(4)
Chapter 11 Sparse sets in the Dynamical Mordell-Lang Conjecture
179(38)
11.1 Overview of the results presented in this chapter
179(3)
11.2 Sets of positive Banach density
182(3)
11.3 General quantitative results
185(4)
11.4 The Dynamical Mordell-Lang problem for Noetherian spaces
189(4)
11.5 Very sparse sets in the Dynamical Mordell-Lang problem for endomorphisms of (P1)N
193(5)
11.6 Reductions in the proof of Theorem 11.5.0.2
198(1)
11.7 Construction of a suitable p-adic analytic function
199(3)
11.8 Conclusion of the proof of Theorem 11.5.0.2
202(3)
11.9 Curves
205(2)
11.10 An analytic counterexample to a p-adic formulation of the Dynamical Mordell-Lang Conjecture
207(2)
11.11 Approximating an orbit by a p-adic analytic function
209(8)
Chapter 12 Denis-Mordell-Lang Conjecture
217(14)
12.1 Denis-Mordell-Lang Conjecture
217(5)
12.2 Preliminaries on function field arithmetic
222(2)
12.3 Proof of our main result
224(7)
Chapter 13 Dynamical Mordell-Lang Conjecture in positive characteristic
231(18)
13.1 The Mordell-Lang Conjecture over fields of positive characteristic
232(1)
13.2 Dynamical Mordell-Lang Conjecture over fields of positive characteristic
233(1)
13.3 Dynamical Mordell-Lang Conjecture for tori in positive characteristic
234(3)
13.4 The Skolem-Mahler-Lech Theorem in positive characteristic
237(12)
Chapter 14 Related problems in arithmetic dynamics
249(14)
14.1 Dynamical Manin-Mumford Conjecture
249(3)
14.2 Unlikely intersections in dynamics
252(2)
14.3 Zhang's conjecture for Zariski dense orbits
254(3)
14.4 Uniform boundedness
257(1)
14.5 Integral points in orbits
258(1)
14.6 Orbits avoiding points modulo primes
259(2)
14.7 A Dynamical Mordell-Lang conjecture for value sets
261(2)
Chapter 15 Future directions
263(4)
15.1 What is known?
263(1)
15.2 What is next?
263(1)
15.3 Varieties with many rational points
264(1)
15.4 A higher dimensional Dynamical Mordell-Lang Conjecture
264(3)
Bibliography 267(10)
Index 277
Jason P. Bell, University of Waterloo, Ontario, Canada.

Dragos Ghioca, University of British Columbia, Vancouver, BC, Canada.

Thomas J. Tucker, University of Rochester, NY, USA.