I recently started teaching a “polynomial method” class, and I’m trying to write detailed lecture notes for it. So far I put the first three chapters online. I’d appreciate any comments about these, from pointing out serious mistakes, to pointing out minor typos, or even just to ask about things that are not so clear. The chapters that are already up are:

- Chapter 1 surveys some classical discrete geometry, and is an introduction to incidences, unit distances, distinct distances, etc.
- Chapter 2 introduces some of the basic real algebraic geometry that will be required in the following chapters. It defines basic concepts such as varieties, dimension, degree, and singular points.
- In chapter 3 we finally start to talk about the polynomial method itself. We introduce the technique of polynomial partitioning and show how to use it to derive incidence bounds.

Chapter 4, about constant-sized polynomial partitioning, is more or less written and should be online soon. I have already received many useful comments from Frank de Zeeuw, who is teaching a similar course in EPFL. Many thanks Frank!

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Good stuff! Chapter 1: In Theorem 4.1, looks like m is undefined, and you end up proving a more general result than stated in the theorem. Also after the thm statement, ‘best know’ –> ‘best known’. Page 7, suggest ‘would be a proof’ –> ‘will be a proof’.

Chap. 2, statement of the Basis Thm. looks trivially true the way you’ve set things up (varieties being defined by finite sets of polys). Chap. 3, Claim 4.1, Gamma-prime looks to be undefined.

Thank you Andy for the nice comments (which I just applied). I appreciate it.

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