The dynamical Mordell-Lang conjecture:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2016]
|
| Series: | Mathematical surveys and monographs
volume 210 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiii, 280 Seiten |
| ISBN: | 9781470424084 |
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MARC
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| 100 | 1 | |a Bell, Jason P. |d 1974- |e Verfasser |0 (DE-588)1106401247 |4 aut | |
| 245 | 1 | 0 | |a The dynamical Mordell-Lang conjecture |c Jason P. Bell, Dragos Ghioca, Thomas J. Tucker |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2016] | |
| 264 | 4 | |c © 2016 | |
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| 490 | 1 | |a Mathematical surveys and monographs |v volume 210 | |
| 650 | 4 | |a Mordell conjecture | |
| 650 | 4 | |a Curves, Algebraic | |
| 650 | 4 | |a Arithmetical algebraic geometry | |
| 650 | 4 | |a Geometry, Algebraic | |
| 700 | 1 | |a Ghioca, Dragos |e Verfasser |0 (DE-588)1106402065 |4 aut | |
| 700 | 1 | |a Tucker, Thomas J. |e Verfasser |0 (DE-588)1106402251 |4 aut | |
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Record in the Search Index
| _version_ | 1804176530850447360 |
|---|---|
| adam_text | Mathematical
Surveys
and
Monographs
Volume 210
The Dynamical
Mordell-Lang
Conjecture
Jason P Bell
Dragos Ghioca
Thomas J Tucker
American Mathematical Society
Providence, Rhode Island
Contents
Preface xi
Notation xii
Chapter 1 Introduction 1
1 1 Overview of the problem 1
1 2 Linear recurrence sequences 2
1 3 Polynomial-exponential Diophantine equations 3
1 4 Linear algebra 4
1 5 Arithmetic geometry 5
1 6 Plan of the book 7
Chapter 2 Background material 11
2 1 Algebraic geometry 11
2 2 Dynamics of endomorphisms 22
2 3 Valuations 24
2 4 Chebotarev Density Theorem 32
2 5 The Skolem-Mahler-Lech Theorem 33
2 6 Heights 40
Chapter 3 The Dynamical Mordell-Lang problem 47
3 1 The Dynamical Mordell-Lang Conjecture 47
3 2 The case of rational self-maps 52
3 3 Known cases of the Dynamical Mordell-Lang Conjecture 54
3 4 The Mordell-Lang conjecture 57
3 5 Denis-Mordell-Lang conjecture 61
36A more general Dynamical Mordell-Lang problem 62
Chapter 4 A geometric Skolem-Mahler-Lech Theorem 67
4 1 Geometric reformulation 67
4 2 Automorphisms of affine varieties 68
4 3 Etale maps 70
4 4 Proof of the Dynamical Mordell-Lang Conjecture for etale maps 73
Chapter 5 Linear relations between points in polynomial orbits 85
5 1 The main results 85
5 2 Intersections of polynomial orbits 89
53A special case 91
5 4 Proof of Theorem 5302 93
5 5 The general case of Theorem 5301 98
5 6 The method of specialization and the proof of Theorem 5502 98
vii
CONTENTS
viii
5 7 The case of Theorem 5201 when the polynomials have different
degrees 102
5 8 An alternative proof for the function field case 107
5 9 Possible extensions 110
5 10 The case of plane curves 110
5 11 A Dynamical Mordell-Lang type question for polarizable
endomor phisms 113
Chapter 6 Parametrization of orbits 117
6 1 Rational maps 118
6 2 Analytic uniformization 120
6 3 Higher dimensional parametrizations 124
Chapter 7 The split case in the Dynamical Mordell-Lang Conjecture 127
7 1 The case of rational maps without periodic critical points 129
7 2 Extension to polynomials with complex coefficients 132
7 3 The case of “almost” post-critically finite rational maps 136
Chapter 8 Heuristics for avoiding ramification 143
81A random model heuristic 143
8 2 Random models and cycle lengths 146
8 3 Random models and avoiding ramification 148
8 4 The case of split maps 150
Chapter 9 Higher dimensional results 153
9 1 The Herman-Yoccoz method for periodic attracting points 153
9 2 The Herman-Yoccoz method for periodic indifferent points 158
9 3 The case of semiabelian varieties 159
9 4 Preliminaries from linear algebra 160
9 5 Proofs for Theorems 9201 and 9301 162
Chapter 10 Additional results towards the Dynamical Mordell-Lang
Conjecture 167
10 1 A v-adic analytic instance of the Dynamical Mordell-Lang Conjecture 167
10 2 A real analytic instance of the Dynamical Mordell-Lang Conjecture 171
10 3 Birational polynomial self-maps on the affine plane 175
Chapter 11 Sparse sets in the Dynamical Mordell-Lang Conjecture 179
11 1 Overview of the results presented in this chapter 179
11 2 Sets of positive Banach density 182
11 3 General quantitative results 185
11 4 The Dynamical Mordell-Lang problem for Noetherian spaces 189
11 5 Very sparse sets in the Dynamical Mordell-Lang problem for
endomorphisms of (P1)^ 193
11 6 Reductions in the proof of Theorem 11 502 198
11 7 Construction of a suitable p-adic analytic function 199
11 8 Conclusion of the proof of Theorem 11 502 202
11 9 Curves 205
11 10 An analytic counterexample to a p-adic formulation of the
Dynamical Mordell-Lang Conjecture 207
CONTENTS
ix
11 11 Approximating an orbit by a p-adic analytic function 209
Chapter 12 Denis-Mordell-Lang Conjecture 217
12 1 Denis-Mordell-Lang Conjecture 217
12 2 Preliminaries on function field arithmetic 222
12 3 Proof of our main result 224
Chapter 13 Dynamical Mordell-Lang Conjecture in positive characteristic 231
13 1 The Mordell-Lang Conjecture over fields of positive characteristic 232
13 2 Dynamical Mordell-Lang Conjecture over fields of positive
characteristic 233
13 3 Dynamical Mordell-Lang Conjecture for tori in positive characteristic 234
13 4 The Skolem-Mahler-Lech Theorem in positive characteristic 237
Chapter 14 Related problems in arithmetic dynamics 249
14 1 Dynamical Manin-Mumford Conjecture 249
14 2 Unlikely intersections in dynamics 252
14 3 Zhang’s conjecture for Zariski dense orbits 254
14 4 Uniform boundedness 257
14 5 Integral points in orbits 258
14 6 Orbits avoiding points modulo primes 259
14 7 A Dynamical Mordell-Lang conjecture for value sets 261
Chapter 15 Future directions 263
15 1 What is known? 263
15 2 What is next? 263
15 3 Varieties with many rational points 264
15 4 A higher dimensional Dynamical Mordell-Lang Conjecture 264
Bibliography 267
|
| any_adam_object | 1 |
| author | Bell, Jason P. 1974- Ghioca, Dragos Tucker, Thomas J. |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV043731430 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:33:39Z |
| institution | BVB |
| isbn | 9781470424084 |
| language | English |
| lccn | 2015036689 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029143288 |
| oclc_num | 950868756 |
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| owner | DE-19 DE-BY-UBM DE-11 |
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| physical | xiii, 280 Seiten |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Bell, Jason P. 1974- Verfasser (DE-588)1106401247 aut The dynamical Mordell-Lang conjecture Jason P. Bell, Dragos Ghioca, Thomas J. Tucker Providence, Rhode Island American Mathematical Society [2016] © 2016 xiii, 280 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs volume 210 Mordell conjecture Curves, Algebraic Arithmetical algebraic geometry Geometry, Algebraic Ghioca, Dragos Verfasser (DE-588)1106402065 aut Tucker, Thomas J. Verfasser (DE-588)1106402251 aut Erscheint auch als Online-Ausgabe 978-1-4704-2908-9 Mathematical surveys and monographs volume 210 (DE-604)BV000018014 210 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029143288&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bell, Jason P. 1974- Ghioca, Dragos Tucker, Thomas J. The dynamical Mordell-Lang conjecture Mathematical surveys and monographs Mordell conjecture Curves, Algebraic Arithmetical algebraic geometry Geometry, Algebraic |
| title | The dynamical Mordell-Lang conjecture |
| title_auth | The dynamical Mordell-Lang conjecture |
| title_exact_search | The dynamical Mordell-Lang conjecture |
| title_full | The dynamical Mordell-Lang conjecture Jason P. Bell, Dragos Ghioca, Thomas J. Tucker |
| title_fullStr | The dynamical Mordell-Lang conjecture Jason P. Bell, Dragos Ghioca, Thomas J. Tucker |
| title_full_unstemmed | The dynamical Mordell-Lang conjecture Jason P. Bell, Dragos Ghioca, Thomas J. Tucker |
| title_short | The dynamical Mordell-Lang conjecture |
| title_sort | the dynamical mordell lang conjecture |
| topic | Mordell conjecture Curves, Algebraic Arithmetical algebraic geometry Geometry, Algebraic |
| topic_facet | Mordell conjecture Curves, Algebraic Arithmetical algebraic geometry Geometry, Algebraic |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029143288&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
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