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Polynomial Methods and Incidence Theory Polynomial Methods and Incidence Theory Adam Sheffer This document is an incomplete draft from July 26, 2020. Several chapters are still missing. Acknowledgements This book would not have been written without Micha Sharir, from whom I learned much of the Discrete Geometry that I know, and Joshua Zahl, from whom I learned much of the Real Algebraic Geometry that I know. I am indebted to Frank de Zeeuw for carefully reading and commenting on earlier versions of this book, and to Nets Katz for being patient with me spending a large amount of time on it. I would like to thank the many mathematicians who helped improving this book. These include Boris Aronov, Abdul Basit, Zachary Chase, Ana Chavez Caliz, Alan Chang, Daniel Di Benedetto, Jordan Ellenberg, Evan Fink, Davey Fitzpatrick, Nora Frankl, Alex Iosevich, Ben Lund, Bob Krueger, Brett Leroux, Shachar Lovett, Michael Manta, Brendan Murphy, Jason O'Neill, Yumeng Ou, Cosmin Pohoata, Piotr Pokora, Anurag Sahay, Steven Senger, Olivine Silier, Shakhar Smorodinsky, Noam Solomon, Samuel Spiro, Jonathan Tidor, and Bartosz Walczak. iii Contents Introduction vii How to read this book............................. ix Notation and inequalities............................x 1 Incidences in Classical Discrete Geometry1 1.1 Introduction................................1 1.2 First proofs................................2 1.3 The crossing lemma............................4 1.4 Szemer´edi-Trotter via the crossing lemma................7 1.5 The unit distances problem.......................8 1.6 The distinct distances problem...................... 10 1.7 A problem about unit area triangles................... 12 1.8 The sum-product problem........................ 13 1.9 Rich points................................ 15 1.10 Exercises.................................. 16 1.11 Open problems.............................. 18 2 Basic Real Algebraic Geometry in R2 19 2.1 Varieties.................................. 19 2.2 Curves in R2 ................................ 21 2.3 Exercises.................................. 25 3 Polynomial Partitioning 26 3.1 The polynomial partitioning theorem.................. 26 3.2 Incidences with algebraic curves in R2 .................. 27 3.3 Proving the polynomial partitioning theorem.............. 32 3.4 Curves containing lattice points..................... 35 3.5 Exercises.................................. 36 iv CONTENTS v 3.6 Open problems.............................. 38 4 Basic Real Algebraic Geometry in Rd 41 4.1 Ideals................................... 41 4.2 Dimension................................. 43 4.3 Singular points.............................. 45 4.4 Degrees.................................. 47 4.5 Polynomial partitioning in Rd ...................... 49 4.6 Exercises.................................. 50 5 The Joints Problem and Degree Reduction 51 5.1 The joints problem............................ 51 5.2 Additional applications of the polynomial argument.......... 54 5.3 (Optional) The probabilistic argument................. 56 5.4 Exercises.................................. 58 6 Polynomial Methods in Finite Fields 59 6.1 Preliminaries............................... 59 6.2 The finite field Kakeya problem..................... 60 6.3 The cap set problem........................... 62 6.4 Warmups: two distances and odd towns................ 63 6.5 The slice rank method.......................... 65 6.6 Exercises.................................. 69 7 Constant-degree Polynomial Partitioning and Incidences in C2 71 7.1 Introduction: Incidence issues in C2 and Rd .............. 71 7.2 Curves in higher dimensions....................... 73 7.3 Constant-degree polynomial partitioning................ 75 7.4 The Szemer´edi{Trotter theorem in C2 .................. 78 7.5 (Optional) Arbitrary curves in C2 .................... 82 7.6 Exercises.................................. 88 7.7 Open problems.............................. 88 8 The Elekes{Sharir{Guth{Katz Framework 90 8.1 Warmup: Distances between points on two lines............ 91 8.2 The ESGK Framework.......................... 94 8.3 (Optional) Lines in the parametric space R3 .............. 98 8.4 Exercises.................................. 99 vi CONTENTS 8.5 Open Problems.............................. 99 9 Lines in R3 101 9.1 From line intersections to incidences................... 101 9.2 Rich points in R3 ............................. 104 9.3 (Optional) Lines in a two-dimensional surface............. 111 9.4 Exercises.................................. 114 9.5 Open problems.............................. 115 10 Distinct Distances Variants 117 10.1 Subsets with no repeated distances................... 117 10.2 Point sets with few distinct distances.................. 120 10.3 Exercises.................................. 121 11 Incidences in Rd 122 11.1 Warmup: Incidences with curves in R3 ................. 122 11.2 Hilbert polynomials............................ 126 11.3 A general point-variety incidence bound................ 128 11.4 Exercises.................................. 135 11.5 Open problems.............................. 136 12 Applications in Rd 138 12.1 Distinct distances with local properties................. 138 12.2 Additive energy on a hypersphere.................... 141 12.3 Exercises.................................. 145 12.4 Open problems.............................. 145 13 Incidences in Spaces Over Finite Fields 147 13.1 Preliminaries............................... 147 13.2 A brief introduction to the projective plane............... 149 13.3 Incidences between large sets of points and lines............ 151 3 13.4 Incidences with planes in Fq ....................... 155 13.5 The sum-product problem in finite fields................ 162 13.6 Incidences between medium-sized sets of points and lines....... 164 13.7 Exercises.................................. 171 13.8 Open problems.............................. 171 Bibliography 173 Introduction \Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine." / Michael Atiyah [3]. In Paul Halmos's famous essay on how to write mathematics [53], Halmos writes \Just as there are two ways for a sequence not to have a limit (no cluster points or too many), there are two ways for a piece of writing not to have a subject (no ideas or too many)." The book you are now starting has two main subjects, which is hopefully a reasonable amount. These two subjects, the polynomial method and incidence theory, are closely tied and hard to separate. Geometric incidences are a family of combinatorial problems, which existed for many decades as part of discrete geometry. In the past decade, incidence problems have been experiencing a renaissance. New interesting connections between incidences and other parts of mathematics are constantly being exposed (such as harmonic anal- ysis, theoretical computer science, model theory, and number theory). At the same time, significant progress is being made on long-standing open incidence problems. The study of geometric incidences is currently an active and exciting research field. One purpose of this book is to survey this field, the recent developments in it, and a variety of connections to other fields. Figure 1: A configuration of four points, four lines, and nine incidences. In an incidence problem we have a set of points and a set of geometric objects . An incidence is a pair (p; V ) such that theP point p is contained in the objectV 2 P × V vii viii CONTENTS V . We denote by I( ; ) the number of incidences in , and (most commonly) wish to study the maximumP V value I( ; ) can have. OneP × Vof the simplest incidence problems studies the maximum numberP V of incidences between m points and n lines in the real plane (see Figure1). Other variants include incidences with other types of curves, incidences with higher-dimensional algebraic objects in Rd, and incidences with semi-algebraic sets in Rd. Incidence problems are also being studied in Cd, in spaces over finite fields, o-minimal structures, and more. Much of the recent progress in studying incidence problems is due to new algebraic techniques. One may describe the philosophy behind these techniques as Collections of objects that exhibit extremal behavior often have hidden algebraic structure. This algebraic structure can be exploited to gain a better understanding of the original problem. For example, in a point-line configuration that determines many incidences, one might expect the point set to have some sort of a lattice structure. Intuitively, one exposes the algebraic structure by defining polynomials according to the problem, and then studying properties of these polynomials. In an incidence problem, one might wish to study a polynomial that vanishes on the point set. This approach is often referred to as the polynomial method. In our study of incidences, we will focus on polynomial methods. In addition, we will see how polynomials methods are used to study problems that do not directly involve incidences. The polynomial approach to studying incidence problems started around 2010. The field is still developing, and in some sense the foundations are not completely established yet. In particular, there are many interesting open problems, some which have not been thoroughly studied yet. Many chapters end by describing such open problems and conjectures. These are mostly long-standing difficult problems,
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