Incidences with Curves in Rd

• Published in: Algorithms - ESA 2015 Vol. 9294; pp. 977 - 988
• Main Authors: Sharir, Micha, Sheffer, Adam, Solomon, Noam
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2015; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Computer programming / software development; Discrete mathematics; Programming & scripting languages: general
• ISBN: 3662483491; 9783662483497
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-662-48350-3_81

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in ℝd is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left.+m+n\right),$\end{document} where the constant of proportionality depends on k, ε and d, for any ε > 0, provided that no j-dimensional surface of degree 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 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.