Zero-patterns of polynomials and Newton polytopes

We present an upper bound on the number of regions into which affine space or the torus over a field may be partitioned by the vanishing and non-vanishing of a finite collection of multivariate polynomials. The bound is related to the number of lattice points in the Newton polytopes of the polynomials, and is optimal to within a factor depending only on the dimension (assuming suitable inequalities hold amongst the relevant parameters). This refines previous work by different authors.