Before revision: In 2015, Guth proved that for any set of k-dimensional  varieties in R^d and for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides R^d into open connected "cells, " so that only a small fraction of the given varieties intersect each cell . Guth's result generalized an earlier result of Guth and Katz for points.
After revision: In 2015, Guth proved that for any set of k-dimensional varieties in R^d and for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides R^d into open connected "cells, " so that only a small fraction of the given varieties intersect each of these sets . Guth's result generalized an earlier result of Guth and Katz for points.