29,873 research outputs found
Incidences with Curves in ℝ^d
We prove that the number of incidences between m points
and n bounded-degree curves with k degrees of freedom in R^d is I(P, C) =
O (mdk-d+1/^k +ε_n^(dk)−d−+1/dk-d + ∑^(d-1)/_(j=2) m/jk−j+1+/k ε_n d(j−1)(k−1)/(d−1)(jk−j+1) q_j /(d−j)(k−1)
(d−1)(jk−j+1) +m + n), 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 [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
Bounds of incidences between points and algebraic curves
We prove new bounds on the number of incidences between points and higher
degree algebraic curves. The key ingredient is an improved initial bound, which
is valid for all fields. Then we apply the polynomial method to obtain global
bounds on and .Comment: 11 page
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane.
Given a set of points in and a set of curves with
degrees of freedom, Pach and Sharir proved that the number of point-curve
incidences is . We
establish the slightly weaker bound
on the number of incidences between points and (complex) algebraic
curves in with degrees of freedom. We combine tools from
algebraic geometry and differential geometry to prove a key technical lemma
that controls the number of complex curves that can be contained inside a real
hypersurface. This lemma may be of independent interest to other researchers
proving incidence theorems over .Comment: The proof was significantly simplified, and now relies on the
Picard-Lindelof theorem, rather than on foliation
Ramsey-type theorems for lines in 3-space
We prove geometric Ramsey-type statements on collections of lines in 3-space.
These statements give guarantees on the size of a clique or an independent set
in (hyper)graphs induced by incidence relations between lines, points, and
reguli in 3-space. Among other things, we prove that: (1) The intersection
graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}).
(2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all
stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no
6-subset is stabbed by one line. (3) Every set of n lines in general position
in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a
subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus.
The proofs of these statements all follow from geometric incidence bounds --
such as the Guth-Katz bound on point-line incidences in R^3 -- combined with
Tur\'an-type results on independent sets in sparse graphs and hypergraphs.
Although similar Ramsey-type statements can be proved using existing generic
algebraic frameworks, the lower bounds we get are much larger than what can be
obtained with these methods. The proofs directly yield polynomial-time
algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi
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