An efficient algorithm for generalized polynomial partitioning and its applications

In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in R^d and if D >= 1 is an integer, then there is a d-variate polynomial P of degree at most D so that each connected component of R^d Z(P) intersects O(n/D^{d-g}) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently - the expected running time of our...