Incidences with Curves in ℝd

  • Micha Sharir
  • Adam Sheffer
  • Noam Solomon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)


We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in ℝ d is Open image in new window \(\left.+m+n\right),\) where the constant of proportionality depends on k, ε and d, for any ε > 0, provided that no j-dimensional surface of degree c j (k,d,ε), a constant parameter depending on k, d, j, and ε, contains more than q j input curves, and that the q j ’s satisfy certain mild conditions.

This bound generalizes a recent result of Sharir and Solomon [20] concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth [8] (as well as the earlier bound of Guth and Katz [10]) in three dimensions (Guth’s three-dimensional bound has a better dependency on q). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl [7], in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi [4] and by Hablicsek and Scherr [11] concerning rich lines in high-dimensional spaces.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Dept. of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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