  Given a set $\Gamma$ 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
$O(D^{k-n} |\Gamma|)$ varieties of $\Gamma$.
