发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为Lewko与Miao–Xie分别提出的两个Szemerédi–Trotter定理推广证明提供几何直观并比较，利用Hasse导数改进证明，最终得到任意代数闭域上的紧关联对数量上界。
AI 中文摘要
Szemerédi–Trotter定理是 incidence geometry（关联几何）中的基础定理，它指出：在ℝ²中，n个点与m条直线构成的关联对数量最多为O(m + n + m^(2/3)n^(2/3))。近期，Lewko以及Miao–Xie分别将Szemerédi–Trotter定理推广到任意域，当域具有正特征𝔭时，需添加修正项O(mn/𝔭)。本文为这两个证明提供了几何直观并对其进行比较，还展示了如何利用Hasse导数改进这两个证明，最终在任意代数闭域上得到紧上界：具体而言，当域具有正特征𝔭，n≤m，且存在正整数k使得𝔭^(2(k-1))≤m<𝔭^(2k)时，关联对数量可被界定为O(𝔭^(k-1)n + m + m^(2/3)n^(2/3) + mn/𝔭^k)。
英文摘要
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 \[O\left(\mathfrak{p}^{k-1}n+m+m^{2/3}n^{2/3}+\frac{mn}{\mathfrak{p}^{k}}\right).\]
Comments25 pages, 3 figures
