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.