Polynomial partitioning for several sets of varieties

TL;DR

This paper presents a new systematic proof extending Guth's polynomial partitioning theorem to multiple families of low-degree varieties in Euclidean space, using topological methods involving equivariant cohomology.

Contribution

It introduces a topologically motivated, unified proof for polynomial partitioning applicable to several sets of varieties, generalizing Guth's original result.

Findings

Extended polynomial partitioning to multiple families of varieties

Used equivariant cohomology and Euler class calculations

Provided bounds on the number of varieties intersected by partition components

Abstract

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.