Polynomial partitioning for several sets of varieties

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 \(D\ge 1\) 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 \(1\le i\le j\). 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.