Polynomial Methods in Combinatorics
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UNIVERSITY LECTURE SERIES VOLUME 64 Polynomial Methods in Combinatorics Larry Guth American Mathematical Society https://doi.org/10.1090//ulect/064 Polynomial Methods in Combinatorics UNIVERSITY LECTURE SERIES VOLUME 64 Polynomial Methods in Combinatorics Larry Guth American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jordan S. Ellenberg Robert Guralnick William P. Minicozzi II (Chair) Tatiana Toro 2010 Mathematics Subject Classification. Primary 05D99. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-64 Library of Congress Cataloging-in-Publication Data Names: Guth, Larry, 1977– Title: Polynomial methods in combinatorics / Larry Guth. Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Univer- sity lecture series ; volume 64 | Includes bibliographical references. Identifiers: LCCN 2016007729 | ISBN 9781470428907 (alk. paper) Subjects: LCSH: Combinatorial geometry. | Polynomials. | Geometry, Algebraic. | AMS: Combina- torics – Extremal combinatorics – None of the above, but in this section. msc Classification: LCC QA167 .G88 2016 | DDC 511/.66–dc23 LC record available at http://lccn.loc. gov/2016007729 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 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 combinatorics 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. Exercises 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. Conclusion 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 Szemer´edi-Trotter theorem 64 7.2. Crossing numbers and the Szemer´edi-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 L(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 Szemer´edi-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. Harmonic 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. Conclusion 267 Bibliography 269 Preface This book explains some recent progress in combinatorial geometry that comes from an unexpected connection with polynomials and algebraic geometry. One of the early results in this story is a two-page solution of a problem called the finite field Kakeya problem, which experts had believed was extremely deep. The most well-known result in this book is an essentially sharp estimate for the distinct distance problem in the plane, a famous problem raised by Paul Erd˝os in the 1940s. The book also emphasizes connections between different fields of mathematics. For example, some of the new proofs in combinatorics that we study were suggested by ideas from error-correcting codes. We discuss this connection, as well as related ideas in Fourier analysis, number theory, and differential geometry. First- or second- year graduate students, as well as advanced undergraduates and researchers, should find this book accessible. My own work in this area is mostly joint with Nets Katz, and I learned a lot about this circle of ideas talking with him and exploring together. I taught a class on this material at MIT in the fall of 2012. I want to thank the students in the class who typed up notes for some of the lectures. Those lecture notes formed a first draft for the book. The students were Sam Elder, Andrey Grinshpun, Nate Harmon, Adam Hesterberg, Chiheon Kim, Gaku Liu, Laszlo Lovasz, Rik Sengupta, Efrat Shaposhnik, Sean Simmons, Yi Sun, Adrian Vladu, Ben Yang, and Yufei Zhao. I also want to thank the following people for looking at drafts of the book and making helpful suggestions: Josh Zahl, Thao Do, Hong Wang, Ben Yang, and Jiri Matou˘sek. While I was writing the book, I was supported by a Sloan fellowship and a Simons Investigator award. Finally, I would like to thank my family for their love and support. Larry Guth, MIT ix Bibliography [AA] P. Agarwal, B. Aronov, Counting facets and incidences, Discrete Comput. Geom. 7 (1992) 359-369. [ACNS] M. Ajtai, V. Chv´atal, M. Newborn, and E. Szemer´edi, Crossing-free subgraphs. Theory and practice of combinatorics, 9-12, North-Holland Math.