Multilevel polynomial partitioning and semialgebraic hypergraphs: regularity, Tur\'an, and Zarankiewicz results

We prove three main results about semialgebraic hypergraphs. First, we prove an optimal and oblivious regularity lemma. Fox, Pach, and Suk proved that the class of $k$-uniform semialgebraic hypergraphs satisfies a very strong regularity lemma where the vertex set can be partitioned into $\mathrm{poly}(1/\varepsilon)$ parts so that all but an $\varepsilon$-fraction of $k$-tuples of parts are homogeneous (either complete or empty). Our result improves the number of parts in the partition to $O_{d,k}((D/\varepsilon)^{d})$ where $d$ is the dimension of the ambient space and $D$ is a measure of the complexity of the hypergraph; additionally, the partition is oblivious to the edge set of the hypergraph. We give examples that show that the dependence on both $\varepsilon$ and $D$ is optimal. From this regularity lemma we deduce the best-known Tur\'an-type result for semialgebraic hypergraphs. Third, we prove a Zarankiewicz-type result for semialgebraic hypergraphs. Previously Fox, Pach, Sheffer, Suk, and Zahl showed that a $K_{u,u}$-free semialgebraic graph on $N$ vertices has at most $O_{d,D,u}(N^{2d/(d+1)+o(1)})$ edges and Do extended this result to $K_{u,\ldots,u}^{(k)}$-free semialgebraic hypergraphs. We improve upon both of these results by removing the $o(1)$ in the exponent and making the dependence on $D$ and $u$ explicit and polynomial. All three of these results follow from a novel ``multilevel polynomial partitioning scheme'' that efficiently partitions a point set $P\subset\mathbb{R}^d$ via low-complexity semialgebraic pieces. We prove this result using the polynomial method over varieties as developed by Walsh which extends the real polynomial partitioning technique of Guth and Katz. We give additional applications to the unit distance problem, the Erd\H{o}s--Hajnal problem for semialgebraic graphs, and property testing of semialgebraic hypergraphs.