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.