Discrete & Computational Geometry

, Volume 48, Issue 2, pp 255–280 | Cite as

An Incidence Theorem in Higher Dimensions



We prove almost tight bounds on the number of incidences between points and k-dimensional varieties of bounded degree in Rd. Our main tools are the polynomial ham sandwich theorem and induction on both the dimension and the number of points.


Szemerédi–Trotter type incidence theorems Sum-product bounds 



The authors are very grateful to Boris Bukh, Nets Katz, Jordan Ellenberg, and Josh Zahl for helpful discussions and to the Isaac Newton Institute, Cambridge for hospitality while this research was being conducted. We also thank Alex Iosevich, Izabella Łaba, Jiří Matoušek, and János Pach for comments, references, and corrections to an earlier draft of this manuscript. We are thankful for the referee for the careful reading and for the suggestions to improve the readability of the paper. The first author is supported by an NSERC grant and the second author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.UCLA Department of MathematicsLos AngelesUSA

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