On lines and Joints

Let L be a set of n lines in R, 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 Θ(n). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [8], and of the follow-up simpler proof of Elekes et al. [5]. Let L be a set of n lines in Rd, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. A simple construction, using the axis-parallel lines in a k × k × · · · × k grid, for k = Θ(n1/(d−1)), has dkd−1 = Θ(n) lines and kd = Θ(nd/(d−1)) joints. In this paper we prove that this is a general upper bound. That is: Theorem 1 The maximum possible number of joints in a set of n lines in Rd is Θ(nd/(d−1)). Background. The problem of bounding the number of joints, for the 3-dimensional case, has been around for almost 20 years [3, 6, 9] (see also [2, Chapter 7.1, Problem 4]), and, until very recently, the best known upper bound, established by Sharir and Feldman [6], was O(n1.6232). The proof techniques were rather complicated, involving a battery of tools from combinatorial geometry, including forbidden subgraphs in extremal graph theory, space decomposition techniques, and some basic results in the geometry of lines in space (e.g., Plücker coordinates). Wolff [10] observed a connection between the problem of counting joints to the Kakeya problem. Bennett et al. [1] exploited this connection and proved an upper bound on the number of so-called θ-transverse joints in R3, namely, joints incident to at least one triple of lines for which the volume of the parallelepiped generated by the three unit vectors along Work on this paper has been partly supported by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work by Micha Sharir has also been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by a grant from the U.S.-Israeli Binational Science Foundation, and by grant 155/05 from the Israel Science Fund. Work by Haim Kaplan has also been supported by Grant 975/06 from the Israel Science Fund, and the United states Israel Binational Science Foundation, project number 2006204. School of Computer Science, Tel Aviv University, Tel Aviv 69978 Israel; [email protected]. School of Computer Science, Tel Aviv University, Tel Aviv 69978 Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA; [email protected]. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978 Israel; [email protected]. these lines is at least θ. This bound is O(n3/2+ε/θ1/2+ε), for any ε > 0, where the constant of proportionality depends on ε. It has long been conjectured that the correct upper bound on the number of joints (in three dimensions) is O(n3/2), matching the lower bound just noted. In a rather dramatic recent development, Guth and Katz [8] have settled the conjecture in the affirmative, showing that the number of joints (in three dimensions) is indeed O(n3/2). Their proof technique is completely different, and uses fairly simple tools from algebraic geometry. In a follow-up paper by Elekes et al. [5], the proof has been further simplified, and extended (a) to obtain bounds on the number of incidences between n lines and (some of) their joints, and (b) to handle also flat points, which are points incident to at least three lines, all coplanar. As far as we know, the problem has not yet been studied for d > 3. In this paper we give a very simple and short proof of Theorem 1; that is, we obtain a tight bound for the maximum possible number of joints in any dimension. The proof uses an algebraic approach similar to that of the other proofs, but is much simpler, shorter and direct. We note that this paper does not subsume the previous paper [5], because the new proof technique cannot handle the problem of counting incidences between lines and joints, nor can it handle flat points. Nevertheless, it is our hope that these extensions would also be amenable to similarly simpler proof techniques. Analysis. We will need the following well-known result from algebraic geometry; see proofs for the 3-dimensional case in [5, 8]. We include the easy general proof for the sake of completeness. Proposition 2 Given a set S of m points in d-space, there exists a nontrivial d-variate polynomial p(x1, . . . , xd) which vanishes at all the points of S, whose degree is at most the smallest integer b satisfying (b+d d )