**University Lecture Series**

Volume: 64;
2016;
273 pp;
Softcover

MSC: Primary 05;

Print ISBN: 978-1-4704-2890-7

Product Code: ULECT/64

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

**Electronic ISBN: 978-1-4704-3214-0
Product Code: ULECT/64.E**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

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#### Supplemental Materials

# Polynomial Methods in Combinatorics

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*Larry Guth*

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

The subject is a beautiful one that has seen contributions by many
leading mathematicians, including the author. The applications of the
polynomial method covered in the book are quite wide-ranging, and run
from combinatorial geometry to diophantine equations.

One particular feature of the book—and of the author's
writing more generally—is the very inviting and discursive
style, emphasising ideas where possible. This is *not* a dry
monograph.

—Ben Joseph Green, Oxford University

Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accomplished using the polynomial method. Larry Guth gives a readable and timely exposition of this important topic, which is destined to influence a variety of critical developments in combinatorics, harmonic analysis and other areas for many years to come.

—Alex Iosevich, University of Rochester, author of “The Erdős Distance Problem” and “A View from the Top”

It is extremely challenging to present a current (and still very active) research area in a manner that a good mathematics undergraduate would be able to grasp after a reasonable effort, but the author is quite successful in this task, and this would be a book of value to both undergraduates and graduates.

—Terence Tao, University of California, Los Angeles, author of “An Epsilon of Room I, II” and “Hilbert's Fifth Problem and Related Topics”

#### Readership

Graduate students and research mathematicians interested in combinatorial incidence geometry, algebraic geometry, and harmonic analysis.

#### Reviews & Endorsements

One of the strengths that combinatorial problems have is that they are understandable to non-experts in the field...One of the strengths that polynomials have is that they are well understood by mathematicians in general. Larry Guth manages to exploit both of those strengths in this book and provide an accessible and enlightening drive through a selection of combinatorial problems for which polynomials have been used to great effect.

-- Simeon Ball, Jahresbericht der Deutschen Mathematiker-Vereinigung

In the 273 page long book, a huge number of concepts are presented, and many results concerning them are formulated and proved. The book is a perfect presentation of the theme.

-- Béla Uhrin, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Polynomial Methods in Combinatorics

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Chapter 1. Introduction 112
- Chapter 2. Fundamental examples of the polynomial method 920
- Chapter 3. Why polynomials? 1930
- Chapter 4. The polynomial method in error-correcting codes 3748
- Chapter 5. On polynomials and linear algebra in combinatorics 5162
- Chapter 6. The Bezout theorem 5566
- Chapter 7. Incidence geometry 6374
- Chapter 8. Incidence geometry in three dimensions 8596
- Chapter 9. Partial symmetries 99110
- 9.1. Partial symmetries of sets in the plane 99110
- 9.2. Distinct distances and partial symmetries 101112
- 9.3. Incidence geometry of curves in the group of rigid motions 103114
- 9.4. Straightening coordinates on 𝐺 104115
- 9.5. Applying incidence geometry of lines to partial symmetries 107118
- 9.6. The lines of \frak𝐿(𝑃) don’t cluster in a low degree surface 108119
- 9.7. Examples of partial symmetries related to planes and reguli 111122
- 9.8. Other exercises 112123

- Chapter 10. Polynomial partitioning 113124
- 10.1. The cutting method 113124
- 10.2. Polynomial partitioning 116127
- 10.3. Proof of polynomial partitioning 117128
- 10.4. Using polynomial partitioning 121132
- 10.5. Exercises 122133
- 10.6. First estimates for lines in \RR³ 126137
- 10.7. An estimate for 𝑟-rich points 128139
- 10.8. The main theorem 129140

- Chapter 11. Combinatorial structure, algebraic structure, and geometric structure 137148
- 11.1. Structure for configurations of lines with many 3-rich points 137148
- 11.2. Algebraic structure and degree reduction 139150
- 11.3. The contagious vanishing argument 140151
- 11.4. Planar clustering 143154
- 11.5. Outline of the proof of planar clustering 144155
- 11.6. Flat points 145156
- 11.7. The proof of the planar clustering theorem 148159
- 11.8. Exercises 149160

- Chapter 12. An incidence bound for lines in three dimensions 151162
- Chapter 13. Ruled surfaces and projection theory 161172
- 13.1. Projection theory 164175
- 13.2. Flecnodes and double flecnodes 172183
- 13.3. A definition of almost everywhere 173184
- 13.4. Constructible conditions are contagious 175186
- 13.5. From local to global 176187
- 13.6. The proof of the main theorem 183194
- 13.7. Remarks on other fields 185196
- 13.8. Remarks on the bound 𝐿^{3/2} 186197
- 13.9. Exercises related to projection theory 187198
- 13.10. Exercises related to differential geometry 189200

- Chapter 14. The polynomial method in differential geometry 195206
- Chapter 15. Harmonic analysis and the Kakeya problem 207218
- 15.1. Geometry of projections and the Sobolev inequality 207218
- 15.2. 𝐿^{𝑝} estimates for linear operators 211222
- 15.3. Intersection patterns of balls in Euclidean space 213224
- 15.4. Intersection patterns of tubes in Euclidean space 218229
- 15.5. Oscillatory integrals and the Kakeya problem 222233
- 15.6. Quantitative bounds for the Kakeya problem 232243
- 15.7. The polynomial method and the Kakeya problem 234245
- 15.8. A joints theorem for tubes 238249
- 15.9. Hermitian varieties 240251

- Chapter 16. The polynomial method in number theory 249260
- 16.1. Naive guesses about diophantine equations 249260
- 16.2. Parabolas, hyperbolas, and high degree curves 251262
- 16.3. Diophantine approximation 254265
- 16.4. Outline of Thue’s proof 258269
- 16.5. Step 1: Parameter counting 259270
- 16.6. Step 2: Taylor approximation 263274
- 16.7. Step 3: Gauss’s lemma 265276
- 16.8. Conclusion 267278

- Bibliography 269280
- Back Cover Back Cover1287