Polynomial Method Lecture Notes #2

Following my previous post, this is an update about my lecture notes for the “polynomial method” class that I am currently teaching. In the last post I briefly described the first three chapters of the lecture notes. Since then, four more chapters are already online and another will be uploaded in the next couple of days. The new chapters are:

  • Chapter 4 describes the constant-degree polynomial partitioning technique, which was introduced by Solymosi and Tao. This technique is useful for deriving incidence bounds in higher dimensions. We prove the complex Szemerédi–Trotter theorem using this technique. (I plan to discuss more recent techniques for handling incidences in higher dimensions in a later chapter.)
  • Chapter 5 consists of a short proof for the joints theorem.
  • Chapter 6 describes the Elekes-Sharir-Guth-Katz framework, which reduces the distinct distances problem to a problem involving line intersections in {\mathbb R}^3  .
  • Chapter 7 revolves around incidences with lines in {\mathbb R}^3  . This is a main step in the proof of Guth and Katz’s distinct distances theorem. The proof of this theorem is completed in Chapter 8.
I am happy to receive corrections and suggestions for improvement. Once again, many thanks to Frank de Zeeuw, for providing a lot of great feedback!