Polynomial partitioning for a set of varieties

TL;DR

This paper proves a polynomial partitioning theorem for low-degree k-dimensional varieties in real space, enabling controlled intersection counts with polynomial zero sets, advancing geometric combinatorics methods.

Contribution

It introduces a polynomial partitioning technique for varieties, extending previous point-based methods to higher-dimensional algebraic sets.

Findings

Existence of a polynomial of degree D partitioning varieties with controlled intersections

Extension of polynomial partitioning to low-degree algebraic varieties

Quantitative bounds on the number of varieties intersecting each partition component

Abstract

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 $\Gamma$.