In their proof of the Erdos distinct distance conjecture in the plane, Guth and Katz developed a "discrete polynomial partitioning" theorem. Given a finite set of points in R^d, this theorem gives a polynomial whose zero-set cuts R^d into "cells," where no cell contains too many points. One difficulty with this method is that many of the points can lie on the zero-set of the partitioning polynomial, and this possibility often complicates whatever analysis one wishes to accomplish. More recent techniques have established a way to perform a second partition on the variety defined by the zero-set of the first partitioning polynomial. To date, however, it has been very difficult to continue this process, and the problem of finding a third (or higher) partitioning polynomial is still open. I will discuss some (limited) progress in this direction.
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