A Szemerédi–Trotter Type Theorem in R[superscript 4]
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Author(s)
Zahl, Joshua
Alternative Title
A Szemerédi–Trotter Type Theorem in R4
Date Issued
August 2015
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Zahl, Joshua. “A Szemerédi–Trotter Type Theorem in $$\mathbb {R}^4$$ R 4.” Discrete Comput Geom 54, no. 3 (August 14, 2015): 513–572.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00454-015-9717-7
