Point-variety incidences, unit distances and Zarankiewicz's problem for algebraic graphs

Abstract: In this paper we study the number of incidences between $m$ points and $n$ varieties in $\mathbb{F}^d$, where $\mathbb{F}$ is an arbitrary field, assuming the incidence graph contains no copy of $K_{s,s}$. We also consider the analogous problem for algebraically defined graphs and unit distance graphs. First, we prove that if $\mathcal{P}$ is a set of $m$ points and $\mathcal{V}$ is a set of $n$ varieties in $\mathbb{F}^{D}$, each of dimension $d$ and degree at most $\Delta$, and in addition the incidence graph is $K_{s,s}$-free, then the number of incidences satisfies $I(\mathcal{P}, \mathcal{V})\leq O_{d,\Delta, s}(m^{\frac{d}{d+1}} n+m)$. This bound is tight when $s,\Delta$ are sufficiently large with respect to $d$, with an appropriate choice of $\mathbb{F}=\mathbb{F}(m,n)$. We give two proofs of this upper bound, one based on the framework of the induced Tur\'an problems and the other based on VC-dimension theory. In the second proof, 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. We also resolve the problem of finding the maximum number of unit distances which can be spanned by a set of $n$ points $\mathcal{P}$ in $\mathbb{F^d}$ whose unit-distance graph is $K_{s, s}$-free, showing that it is $\Theta_{d,s}(n^{2-\frac{1}{\lceil d/2\rceil +1}})$. Finally, we obtain tight bounds on the maximum number of edges of a $K_{s, s}$-free algebraic graph defined over a finite field, thus resolving the Zarankiewicz problem for this class of graphs.