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.

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 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.