On Lines and Joints

Abstract

Let L be a set of n lines in ^sup d^, for d>=3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is [Theta](n ^sup d/(d-1)^). For d=3, this is a considerable simplification of the original algebraic proof of Guth and Katz (Algebraic methods in discrete analogs of the Kakeya problem, 4 December 2008 , arXiv:0812.1043), and of the follow-up simpler proof of Elekes et al. (On lines, joints, and incidences in three dimensions. Manuscript, 11 May 2009 , arXiv:0905.1583). Some extensions, e.g., to the case of joints of algebraic curves, are also presented.[PUBLICATION ABSTRACT]