Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

Abstract

We study a wide spectrum of incidence problems involving points and curves or points and surfaces in R 3 . The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [38,39], requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies [40], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for numerous incidence problems in R 3 .In broad terms, we consider two kinds of problems, those involving points and constant-degree algebraic curves, and those involving points and constant-degree algebraic surfaces. In some variants we assume that the points lie on some fixed constant-degree algebraic variety, and in others we consider arbitrary sets of points in 3-space.The case of points and curves has been considered in several previous studies, starting with Guth and Katz's work on points and lines [39]. Our results, which are based on a recent work of Guth and Zahl [40] concerning surfaces that are doubly ruled by curves, provide a grand generalization of all previous results. We reconstruct the bound for points and lines, and improve, in certain signifcant ways, recent bounds involving points and circles (in [54]), and points and arbitrary constant-degree algebraic curves (in [53]). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge.In the case of points and surfaces, the incidence graph between them can contain large complete bipartite graphs, each involving points on some curve and surfaces containing this curve (unlike earlier studies, we do not rule out this possibility, which makes our approach more general). Our bounds estimate the total size of the vertex sets in such a complete bipartite graph decomposition of the incidence graph. In favorable cases, our bounds translate into actual incidence bounds. Overall, here too our results provide a "grand generalization" of most of the previous studies of (special instances of) this problem.As an application of our point-curve incidence bound, we consider the problem of bounding the number of similar triangles spanned by a set of n points in R 3 . We obtain the bound O(n 15/7 ), which improves the bound of Agarwal et al. [1].As applications of our point-surface incidence bounds, we consider the problems of distinct and repeated distances determined by a set of n points in R 3 , two of the most celebrated open problems in combinatorial geometry. We obtain new and improved bounds for two special cases, one in which the points lie on some algebraic variety of constant degree, and one involving incidences between pairs in P 1 × P 2 , where P 1 is contained in a variety and P 2 is arbitrary.Recently, Sharir and Zahl [60] have considered general families of con...

Representation Complexities of SemiAlgebraic Graphs

“…when P is a set of m points in R d 1 , Q is a set of n points in R d 2 and the edges are defined by some semi-algebraic relations, the representation complexity of G is O(mfor arbitrarily small positive ε. This generalizes results by Apfelbaum-Sharir [4] and Solomon-Sharir [25][26][27]. As a consequence, when G is Ku,u-free for some positive integer u, its number of edges isThis bound is stronger than that of Fox, Pach, Sheffer, Suk and Zahl in [15] when the first term dominates and u grows with m, n. Another consequence is that we can find a large complete bipartite subgraph in a semi-algebraic graph when the number of edges is large.…”

“…for arbitrarily small positive ε. This generalizes results by Apfelbaum-Sharir [4] and Solomon-Sharir [25][26][27]. As a consequence, when G is Ku,u-free for some positive integer u, its number of edges is…”

“…• As far as the author is aware, this is the first result on representation complexity of semi-algebraic sets; previous results in [4,5,[25][26][27] only apply to algebraic sets. • Letting P and Q be sets of points and hyperplanes in R d (so d 1 = d 2 = d), we recover Theorem 1.1 within an ε term.…”

“…• To recover Theorem 1.2, let P be the set of points in V (so e 1 = 2) and Q be the set of surfaces in R 3 . If all surfaces of Q have degrees at most t, Q live in R ( t+3 3 ) , but since these surfaces have k degrees of freedom with respect to V , we expect e 2 = k. The relationship between the degree of freedom and the dimension of the moduli space of a family of curves/surfaces is quite complicated, see [27] and the appendix of [28] for more details. But typically we should expect e 2 = k and hence recover the bound in Theorem 1.2 within an ε term.…”

Incidences between points and curves with almost two degrees of freedom

“…A similar argument can be made for point-circle incidences in R 3 (or again in any dimension ≥ 3)here we need to constrain the number of input circles that can lie in any common plane or sphere. The best known upper bound, due to Sharir and Solomon [12], is (see also Sharir et al [11] for an earlier, weaker bound).…”