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Discrete & Computational Geometry

, Volume 54, Issue 3, pp 513–572 | Cite as

A Szemerédi–Trotter Type Theorem in \(\mathbb {R}^4\)

  • Joshua ZahlEmail author
Article

Abstract

We show that m points and n two-dimensional algebraic surfaces in \({\mathbb {R}}^4\) can have at most \(O(m^{{k}/({2k-1})}n^{({2k-2})/({2k-1})}+m+n)\) incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that \(m\le n^{(2k+2)/3k}\). As a special case, we obtain a Szemerédi–Trotter type theorem for 2-planes in \({\mathbb {R}}^4\), provided \(m\le n\) and the planes intersect transversely. As a further special case, we obtain a Szemerédi–Trotter type theorem for complex lines in \({\mathbb {C}}^2\) 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 \({\mathbb {C}}^2\). We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.

Keywords

Incidence geometry Combinatorial geometry Polynomial partitioning Crossing lemma 

Mathematics Subject Classification

52C35 52C10 32S22 

Notes

Acknowledgments

The author is very grateful to Saugata Basu, Kiran Kedlaya, Silas Richelson, Terence Tao, and Burt Totaro for helpful discussions. The author would like to especially thank the anonymous referees for their careful reading and numerous suggestions. Referee #1 in particular went above and beyond the usual refereeing process, and the author is very grateful for the time and effort he or she put in. The author was supported in part by the Department of Defense through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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