Incidences with Curves in ℝd
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  concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth  (as well as the earlier bound of Guth and Katz ) 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 , in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi  and by Hablicsek and Scherr  concerning rich lines in high-dimensional spaces.
Unable to display preview. Download preview PDF.
- 2.Basu, S., Sombra, M.: Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions. arXiv:1406.2144Google Scholar
- 4.Dvir, Z., Gopi, S.: On the number of rich lines in truly high dimensional sets. In: Proc. 31st Annu. Sympos. Comput. Geom. (2015, to appear)Google Scholar
- 5.Elekes, G., Kaplan, H., Sharir, M.: On lines, joints, and incidences in three dimensions. J. Combinat. Theory, Ser. A 118, 962–977 (2011); Also in arXiv:0905.1583Google Scholar
- 7.Fox, J., Pach, J., Sheffer, A., Suk, A., Zahl, J.: A semi-algebraic version of Zarankiewicz’s problem. arXiv:1407.5705Google Scholar
- 8.Guth, L.: Distinct distance estimates and low-degree polynomial partitioning. Discrete Comput. Geom. 53, 428–444 (2015); Also in arXiv:1404.2321Google Scholar
- 9.Guth, L., Katz, N.H.: Algebraic methods in discrete analogs of the Kakeya problem. Advances Math. 225, 2828–2839 (2010); Also in arXiv:0812.1043Google Scholar
- 10.Guth, L., Katz, N.H.: On the Erdős distinct distances problem in the plane. Annals Math. 181, 155–190 (2015); Also in arXiv:1011.4105Google Scholar
- 11.Hablicsek, M., Scherr, Z.: On the number of rich lines in high dimensional real vector spaces. arXiv:1412.7025Google Scholar
- 13.Kaplan, H., Matoušek, J., Safernová, Z., Sharir, M.: Unit distances in three dimensions. Combinat. Probab. Comput. 21, 597–610 (2012); Also in arXiv:1107.1077Google Scholar
- 14.Kaplan, H., Matoušek, J., Sharir, M.: Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique. Discrete Comput. Geom. 48, 499–517 (2012); Also in arXiv:1102.5391Google Scholar
- 16.Nilov, F., Skopenkov, M.: A surface containing a line and a circle through each point is a quadric. arXiv:1110:2338Google Scholar
- 18.Pach, J., Sharir, M.: Geometric incidences. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 185–223. Amer. Math. Soc., Providence (2004)Google Scholar
- 19.Sharir, M., Sheffer, A., Zahl, J.: Improved bounds for incidences between points and circles. Combinat. Probab. Comput. 24, 490–520 (2015); Also in Proc. 29th ACM Symp. on Computational Geometry, 97–106, arXiv:1208.0053 (2013)Google Scholar
- 20.Sharir, M., Solomon, N.: Incidences between points and lines in four dimensions. In: Proc. 30th ACM Sympos. on Computational Geometry, pp. 189–197 (2014)Google Scholar
- 21.Sharir, M., Solomon, N.: Incidences between points and lines in four dimensions. J. AMS, arXiv:1411.0777 (submitted)Google Scholar
- 22.Sharir, M., Solomon, N.: Incidences between points and lines in three dimensions. In: Proc. 31st ACM Sympos. on Computational Geometry (2015, to appear); Also Discrete Comput. Geom. arXiv:1501.02544 (submitted)Google Scholar
- 23.Sharir, M., Solomon, N.: Incidences between points and lines on a two-dimensional variety in three dimensions. Combinatorica, arXiv:1501.01670 (submitted)Google Scholar
- 25.Spencer, J., Szemerédi, E., Trotter, W.T.: Unit distances in the Euclidean plane. In: Bollobás, B. (ed.) Graph Theory and Combinatorics, pp. 293–303. Academic Press, New York (1984)Google Scholar
- 29.Zahl, J.: An improved bound on the number of point-surface incidences in three dimensions. Contrib. Discrete Math. 8(1) (2013); Also in arXiv:1104.4987Google Scholar
- 30.Zahl, J.: A Szemerédi-Trotter type theorem in ℝ4, arXiv:1203.4600Google Scholar