Polynomial methods in combinatorics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2016]
|
| Schriftenreihe: | University lecture series / American Mathematical Society
volume 64 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | ix, 273 Seiten Illustrationen |
| ISBN: | 9781470428907 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804176424135819264 |
|---|---|
| adam_text | Titel: Polynomial methods in combinatorics
Autor: Guth, Larry
Jahr: 2016
Contents
Preface ix
Chapter 1. Introduction 1
1.1. Incidence geometry 2
1.2. Connections with other areas 4
1.3. Outline of the book 6
1.4. Other connections between polynomials and combinatorios 7
1.5. Notation 7
Chapter 2. Fundamental examples of the polynomial method 9
2.1. Parameter counting arguments 9
2.2. The vanishing lemma 10
2.3. The finite-field Nikodym problem 11
2.4. The finite field Kakeya problem 12
2.5. The joints problem 13
2.6. Comments on the method 15
2.7. Exer eises 17
Chapter 3. Why polynomials? 19
3.1. Finite field Kakeya without polynomials 19
3.2. The Hermitian variety 22
3.3. Joints without polynomials 27
3.4. What is special about polynomials? 32
3.5. An example involving polynomials 33
3.6. Combinatorial structure and algebraic structure 34
Chapter 4. The polynomial method in error-correcting codes 37
4.1. The Berlekamp-Welch algorithm 37
4.2. Correcting polynomials from overwhelmingly corrupted data 40
4.3. Locally decodable codes 41
4.4. Error-correcting codes and finite-field Nikodym 44
4.5. Conclusión and exercises 45
Chapter 5. On polynomials and linear algebra in combinatorics 51
Chapter 6. The Bezout theorem 55
6.1. Proof of the Bezout theorem 55
6.2. A Bezout theorem about surfaces and lines 58
6.3. Hilbert polynomials 60
V
vi
CONTENTS
Chapter 7. Incidence geometry 63
7.1. The Szemerédi-Trotter theorem 64
7.2. Crossing numbers and the Szemerédi-Trotter theorem 67
7.3. The language of incidences 71
7.4. Distance problems in incidence geometry 75
7.5. Open questions 76
7.6. Crossing numbers and distance problems 79
Chapter 8. Incidence geometry in three dimensions 85
8.1. Main results about lines in R3 85
8.2. Higher dimensions 88
8.3. The Zarankiewicz problem 90
8.4. Reguli 95
Chapter 9. Partial symmetries 99
9.1. Partial symmetries of sets in the plane 99
9.2. Distinct distances and partial symmetries 101
9.3. Incidence geometry of curves in the group of rigid motions 103
9.4. Straightening coordinates on G 104
9.5. Applying incidence geometry of lines to partial symmetries 107
9.6. The lines of £(P) don t cluster in a low degree surface 108
9.7. Examples of partial symmetries related to planes and reguli 111
9.8. Other exercises 112
Chapter 10. Polynomial partitioning 113
10.1. The cutting method 113
10.2. Polynomial partitioning 116
10.3. Proof of polynomial partitioning 117
10.4. Using polynomial partitioning 121
10.5. Exercises 122
10.6. First estimates for lines in R3 126
10.7. An estimate for r-rich points 128
10.8. The main theorem 129
Chapter 11. Combinatorial structure, algebraic structure,
and geometric structure 137
11.1. Structure for configurations of lines with many 3-rich points 137
11.2. Algebraic structure and degree reduction 139
11.3. The contagious vanishing argument 140
11.4. Planar clustering 143
11.5. Outline of the proof of planar clustering 144
11.6. Flat points 145
11.7. The proof of the planar clustering theorem 148
11.8. Exercises 149
Chapter 12. An incidence bound for lines in three dimensions 151
12.1. Warmup: The Szemerédi-Trotter theorem revisited 152
12.2. Three-dimensional incidence estimates 154
CONTENTS vii
Chapter 13. Ruled surfaces and projection theory 161
13.1. Projection theory 164
13.2. Flecnodes and double flecnodes 172
13.3. A definition of almost everywhere 173
13.4. Constructible conditions are contagious 175
13.5. From local to global 176
13.6. The proof of the main theorem 183
13.7. Remarks on other fields 185
13.8. Remarks on the bound L3/2 186
13.9. Exercises related to projection theory 187
13.10. Exercises related to differential geometry 189
Chapter 14. The polynomial method in differential geometry 195
14.1. The efficiency of complex polynomials 195
14.2. The efficiency of real polynomials 197
14.3. The Crofton formula in integral geometry 198
14.4. Finding functions with large zero sets 200
14.5. An application of the polynomial method in geometry 201
Chapter 15. Harmonie analysis and the Kakeya problem 207
15.1. Geometry of projections and the Sobolev inequality 207
15.2. Lp estimates for linear operators 211
15.3. Intersection patterns of balls in Euclidean space 213
15.4. Intersection patterns of tubes in Euclidean space 218
15.5. Oscillatory integrals and the Kakeya problem 222
15.6. Quantitative bounds for the Kakeya problem 232
15.7. The polynomial method and the Kakeya problem 234
15.8. A joints theorem for tubes 238
15.9. Hermitian varieties 240
Chapter 16. The polynomial method in number theory 249
16.1. Naive guesses about diophantine equations 249
16.2. Parabolas, hyperbolas, and high degree curves 251
16.3. Diophantine approximation 254
16.4. Outline of Thue s proof 258
16.5. Step 1: Parameter counting 259
16.6. Step 2: Taylor approximation 263
16.7. Step 3: Gauss s lemma 265
16.8. Conclusión 267
Bibliography 269
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| spelling | Guth, Larry 1977- Verfasser (DE-588)1106273656 aut Polynomial methods in combinatorics Larry Guth Providence, Rhode Island American Mathematical Society [2016] © 2016 ix, 273 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier University lecture series / American Mathematical Society volume 64 Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Kombinatorik (DE-588)4031824-2 gnd rswk-swf Polynom (DE-588)4046711-9 gnd rswk-swf Kombinatorik (DE-588)4031824-2 s Polynom (DE-588)4046711-9 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-3214-0 American Mathematical Society University lecture series volume 64 (DE-604)BV004153846 64 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029079479&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Guth, Larry 1977- Polynomial methods in combinatorics Algebraische Geometrie (DE-588)4001161-6 gnd Kombinatorik (DE-588)4031824-2 gnd Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4031824-2 (DE-588)4046711-9 |
| title | Polynomial methods in combinatorics |
| title_auth | Polynomial methods in combinatorics |
| title_exact_search | Polynomial methods in combinatorics |
| title_full | Polynomial methods in combinatorics Larry Guth |
| title_fullStr | Polynomial methods in combinatorics Larry Guth |
| title_full_unstemmed | Polynomial methods in combinatorics Larry Guth |
| title_short | Polynomial methods in combinatorics |
| title_sort | polynomial methods in combinatorics |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd Kombinatorik (DE-588)4031824-2 gnd Polynom (DE-588)4046711-9 gnd |
| topic_facet | Algebraische Geometrie Kombinatorik Polynom |
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