Incidences with curves in R d ∗

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in Rd is O m k dk−d+1n dk−d dk−d+1 + d−1 ∑ j=2 m k jk−j+1n d(j−1)(k−1) (d−1)(jk−j+1) q (d−j)(k−1) (d−1)(jk−j+1) j + m + n  , for any ε > 0, where the constant of proportionality depends on k, ε and d, provided that no j-dimensional surface of degree 6 cj(k, d, ε), a constant parameter depending on k, d, j, and ε, contains more than qj input curves, and that the qj ’s satisfy certain mild conditions. This bound generalizes the well-known planar incidence bound of Pach and Sharir to Rd. It generalizes a recent result of Sharir and Solomon [21] concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth [9] (as well as the earlier bound of Guth and Katz [11]) in three dimensions (Guth’s three-dimensional bound has a better dependency on q2). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl [8], in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi [5] and by Hablicsek and Scherr [13] concerning rich lines in high-dimensional spaces. Our bound is not known to be tight in most cases. ∗Work on this paper was partially supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM) at UCLA, which is supported by the National Science Foundation. A preliminary version of this paper appeared in Proc. European Sympos. Algorithms, 2015, pages 977–988. †Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel; [email protected] ‡Dept. of Mathematics, California Institute of Technology, Pasadena, CA, USA; [email protected] §Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel; [email protected] the electronic journal of combinatorics 23(4) (2016), #P4.16 1