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.

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.