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.

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.

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 Chapter 11 we derived general point-variety incidence bounds in R^d. We now study two applications of these bounds. These applications do not require reading any part of Chapter 11, except for the statement of Theorem 11.3. The first application comes from discrete geometry and is another distinct distances problem. Specifically, it is a distinct distances problem with local properties. The second application is a discrete Fourier restriction problem from harmonic analysis. For simplicity, we only discuss the combinatorial aspect of that problem. This aspect is the additive energy of points on a hypersphere.