A Szemeredi-Trotter type theorem in $\mathbb{R}^4$

TL;DR

This paper extends Szemerédi-Trotter type incidence bounds to points and algebraic surfaces in four-dimensional space, using advanced combinatorial and algebraic tools, with special cases including complex lines and circles.

Contribution

It introduces a new incidence bound for points and algebraic surfaces in , generalizing previous results and applying novel polynomial partitioning and crossing lemma techniques.

Findings

Established an incidence bound for points and algebraic surfaces in .

Derived special cases for 2-planes in  and complex lines in .

Extended Szemerédi-Trotter type theorems to new geometric configurations.

Abstract

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.