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In this chapter, we study general incidence bounds in R^d. As a warm-up, we first derive an incidence bound for curves in R^3. The main result of this chapter is a general point-variety incidence bound in R^d. This result relies on another polynomial partitioning theorem, for the case where the points are on a constant-degree variety. The proof of this partitioning theorem relies on Hilbert polynomials. In particular, we use Hilbert polynomials to derive a polynomial ham sandwich theorem for points on a variety.
This chapter consists of basic real algebraic geometry. In Chapter 2 we studied basic properties of curves in the plane. We now generalize these properties to arbitrary varieties in any real space and introduce additional properties.
We begin the chapter by briefly discussing polynomial ideals. We then study the dimension of a real variety, connected components, irreducible components, tangent spaces, singular points, regular points, and other properties. We discuss why there is no one well-behaved definition for the degree of a real variety. We also study the projection of real varieties and partitioning polynomials in higher dimensions.
We begin our study of geometric incidences by surveying the field and deriving a few first bounds. In this chapter we only discuss classical discrete geometry, from before the discovery of the new polynomial methods. This makes the current chapter rather different than the rest of the book (outrageously, it even includes some graph theory). We also learn basic tricks that are used throughout the book, such as double counting, applying the Cauchy–Schwarz inequality, and dyadic decomposition.
Topics that are discussed in this chapter: the Szemerédi–Trotter theorem, a proof of this theorem that relies on the crossing lemma, the unit distances problem, the distinct distances problem, a problem about unit area triangles, the sum-product problem, rich point, point-line duality.
In Chapter 6 we saw several applications of polynomial methods in finite fields. In this chapter, we continue our study of finite fields, by studying point-line incidences in a finite plane. Much less is known about incidences over finite fields and many incidence problems become more difficult in this case. Unlike incidences over the reals, there is no one main technique that leads to most of the current best bounds. Instead, each bound that we derive in this chapter requires a rather different set of tools.
For the proofs in this chapter, we introduce tools such as the projective plane, eigenvalues of a graph, and more. We also use finite field incidence bounds to study the finite field sum-product problem.
After seeing some basics of incidence theory, we wish to discuss how polynomial methods are used to study incidences. For that, we first need a basic introduction to algebraic geometry over the reals. In this chapter we focus mainly on the plane, postponing the treatment of higher dimensions to Chapter 4. This allows us to discuss several planar results in Chapter 3, before dealing with more involved algebraic geometry.
We begin the chapter by introducing varieties and their properties. We then focus on curves in the plane and their properties: degree, irreducible components, connected components, intersections, and more. We conclude the chapter with a polynomial proof of Pascal’s theorem, due to Plücker.
We started our study of the distinct distances problem in Section 1.6. The mathematicians Elekes and Sharir used to discuss this problem. Around the turn of the millennium, Elekes discovered a reduction from this problem to a problem about intersections of helices in R^3. Elekes said that, if something happens to him, then Sharir should publish their ideas.
Elekes passed away in 2008 and, as requested, Sharir then published their ideas. Before publishing, Sharir simplified the reduction so that it led to a problem about intersections of parabolas in R^3. Sharing the reduction with the general community had surprising consequences. Hardly any time had passed before Guth and Katz managed to apply the reduction to almost completely solve the distinct distances problem.
In this chapter we study the reduction of Elekes, Sharir, Guth, and Katz. This reduction is based on parameterizing rotations of the plane as points in R^3. As a warmup, we begin with a problem about distinct distances between two lines.
In this chapter, we use polynomial methods to study incidence-related problems in spaces over finite fields. We focus on two breakthroughs: A solution to the finite field Kakeya problem and the cap set problem. The proofs of these results are short, elegant, and require mostly elementary tools. In Chapter 13, we study point-line incidences in spaces over finite fields, which require more involved arguments.
This chapter contains a variety of other interesting problems and tools. We study the method of multiplicities, which improves the constant of the finite-field Kakeya theorem. To study the cap set problem, we use the slice rank technique. This technique is also used to obtain bounds for the 3-sunflower problem. As a warm-up towards the slice rank technique, we consider the Odd town problem and the two distance problem.
Before their seminal distinct distances paper, Guth and Katz wrote another paper that introduced a new polynomial method. In this chapter, we study one of the two problems that were resolved in that paper: the joints problem. The solution to this problem relies on a simple polynomial technique, which is based on polynomial interpolation. This is also a good warm-up for working in spaces of dimension larger than two.
We use the polynomial interpolation technique to study two additional problems. First, we study the sets in R^3 that are formed by the union of all lines that intersect three pairwise-skew lines. We then use the degree reduction technique to study polynomial interpolation of lines.
It is usually easier to study problems over the complex than over the reals. Discrete geometry problems are an exception, often being significantly simpler over the reals. While there are several simple proofs of the Szemerédi–Trotter theorem over the reals, we only have rather involved proofs for the complex variant of the theorem. To avoid such involved proofs, we prove a slightly weaker variant of the complex Szemerédi–Trotter theorem. Our analysis is based on thinking of C^2 as R^4.
In Chapter 7, we began to prove the distinct distances theorem by studying the ESGK framework. We complete this proof in Chapter 9, by relying on the constant-degree polynomial partitioning technique. In the current chapter we introduce this technique by studying incidences with lines in the complex plane. This is a warm-up towards Chapter 9, where we use constant-degree polynomial partitioning in more involved ways.
In this chapter, we study our first new polynomial technique: polynomial partitioning. We first see the polynomial partitioning theorem. We use this theorem to derive an incidence bound between points and curves in the real plane. This bound generalizes the Szemerédi–Trotter theorem and the current best bound for the unit distances problem. In the second part of the chapter, we prove the polynomial partitioning theorem by using the ham sandwich theorem and Veronese maps. Finally, we use the point-curve incidence bound to obtain an upper bound for the number of lattice points that a curve can contain.
During the chapter we learn other important concepts, such as Warren’s theorem, incidence graphs, and various tricks for working with curves.
In this chapter we discuss advanced tools and techniques, which rely on additional concepts from algebraic geometry. These tools could be helpful for people who do research work in incidence theory and related topics. A reader who is new to this field might prefer to skip this chapter.
We sometimes wish to consider families of varieties, such as the set of circles in the plane or the set of planes in R^3 that not are incident to the origin. In this chapter, we rigorously define such families. We also generalize the idea of point-line duality to every family of varieties. We then see how these notions could be used to prove various results. In particular, we derive a new incidence bound and prove various properties of surfaces in R^3 and C^3.
After the long and technical proof of the distinct distances theorem, we move to a lighter chapter. In this chapter we study two additional distinct distances problems. We first show that every planar point set contains a large subset that does not span any distance more than once. We then study the structural distinct distances problem: characterizing the point sets that span a small number of distinct distances.
We also study a problem that does not involve distinct distances, but relies on a variant of Theorem 9.2. This problem considers sets of intervals in the plane that span many trapezoids.
In Chapter 7 we studied the ESGK framework. This was a reduction from the distinct distances problem to a problem about pairs of intersecting lines in R^3. In the current chapter we further reduce the problem to bounding the number of rich points of lines in R^3. We solve this incidence problem with a more involved variant of the constant-degree polynomial partitioning technique. This completes the proof of the Guth–Katz distinct distances theorem.
The original proof of Guth and Katz is quite involved. We study a simpler proof for a slightly weaker variant of the distinct distances theorem. This simpler proof was introduced by Guth and avoids the use of tools such as flat points and properties of ruled surfaces.