A Szemerédi–Trotter Type Theorem in R[superscript 4]

Abstract

We show that m points and n two-dimensional algebraic surfaces in R[superscript 4] can have at most O(m[superscript k/(2k−1)n(2k−2)/(2k−1)]+m+n) incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that m≤n[superscript (2k+2)/3k]. As a special case, we obtain a Szemerédi–Trotter type theorem for 2-planes in R[superscript 4], provided m≤n and the planes intersect transversely. As a further special case, we obtain a Szemerédi–Trotter type theorem for complex lines in C[superscript 2] with no restrictions on m and n (this theorem was originally proved by Tóth using a different method). As a third special case, we obtain a Szemerédi–Trotter type theorem for complex unit circles in C2. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.