Before revision: Guth showed that given a family \mathcal{S of n g-dimensional semi-algebraic sets in R^d and an integer parameter D\geq 1 , there is a d-variate partitioning polynomial P of degree at most D , so that each connected component of R^d\setminus Z(P)
After revision: Guth showed that given a family \mathcal{S of n g-dimensional semi-algebraic sets in R^d and if D\geq 1 , there is a d-variate partitioning polynomial P of degree at most D , so that each connected component of R^d\setminus Z(P)