Polynomial partitioning for a set of varieties

Given a set $\mathrm{\Gamma }$ of low-degree k-dimensional varieties in ${\mathbb{R}}^n$, we prove that for any $D\ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of ${\mathbb{R}}^n\setminus Z(P)$ intersects $O(D^{k-n}\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid })$ varieties of $\mathrm{\Gamma }$.