  We give a new, systematic proof for a recent result of Larry Guth and thus
also extend the result to a setting with several families of varieties: For any
integer $D\geq 1$ and any collection of sets $\Gamma_1,\ldots,\Gamma_j$ of
low-degree $k$-dimensional varieties in $\mathbb{R}^n$ there exists a non-zero
polynomial $p\in\mathbb{R}[X_1,\ldots,X_n]$ of degree at most $D$ so that each
connected component of $\mathbb{R}^n{\setminus}Z(p)$ intersects
$O(jD^{k-n}|\Gamma_i|)$ varieties of $\Gamma_i$, simultaneously for every
$1\leq i\leq j$. For $j=1$ we recover the original result by Guth. Our proof,
via an index calculation in equivariant cohomology, shows how the degrees of
the polynomials used for partitioning are dictated by the topology, namely by
the Euler class being given in terms of a top Dickson polynomial.
