Two proofs and a tight Szemerédi--Trotter theorem

Source abstract

The Szemerédi--Trotter theorem, a fundamental theorem in incidence geometry, states that $n$ points and $m$ lines in $\mathbb{R}^2$ form at most $O(m+n+m^{2/3}n^{2/3})$ incidences. Recently, Lewko and independently Miao--Xie extended the Szemerédi--Trotter theorem to arbitrary fields with a correction term $O(\frac{mn}{\mathfrak{p}})$ if the field has positive characteristic $\mathfrak{p}$. In this paper, we provide some geometric intuition for the two proofs and compare them. Moreover, we show how both proofs can be improved using Hasse derivatives. As a consequence, we obtain a tight upper bound over any algebraically closed field. To be specific, when the field has positive characteristic $\mathfrak{p}$, $n\leq m$ and $\mathfrak{p}^{2(k-1)}\leq m<\mathfrak{p}^{2k}$ for some positive integer $k$, the number of incidences can be bounded by $$O\left(\mathfrak{p}^{k-1}n+m+m^{2/3}n^{2/3}+\frac{mn}{\mathfrak{p}^{k}}\right).$$