Polynomial partitioning for a set of varieties

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 $\Gamma$.

11 pages. Revised to make the topological argument more accessible. Accepted for publication in Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (2)

• On the Erdos distinct distance problem in the plane
• Improved bounds for incidences between points and circles

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