Incidence bounds via extremal graph theory

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $$O_{d, s}((mn)^{1-1/(d+1)}+m+n).$$ This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers. In this paper, we propose a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$, e.g. when $m=n$ and $d$ is odd. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. We also study the size of the largest complete bipartite graph in point-hyperplane incidence graphs with a given number of edges and obtain optimal bounds as well. Additionally, we study point-variety incidences and unit-distance problem in finite fields, and give tight bounds for both problems under a similar non-degeneracy assumption. We also resolve Zarankiewicz type problems for algebraic graphs. Our proofs use tools such as induced Tur\'an problems, VC-dimension theory, evasive sets and Hilbert polynomials. Also, we extend the celebrated result of R\'onyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest.