24,099 research outputs found
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
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 m points in R2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is (Formula presented.). We establish the slightly weaker bound (Formula presented.) on the number of incidences between m points and n (complex) algebraic curves in C2 with k 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 C. © 2017 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelber
On Rich Points and Incidences with Restricted Sets of Lines in 3-Space
Let be a set of lines in that is contained, when represented as
points in the four-dimensional Pl\"ucker space of lines in , in an
irreducible variety of constant degree which is \emph{non-degenerate} with
respect to (see below). We show:
\medskip \noindent{\bf (1)} If is two-dimensional, the number of -rich
points (points incident to at least lines of ) is
, for and for any , and, if at
most lines of lie on any common regulus, there are at most
-rich points. For larger than some sufficiently
large constant, the number of -rich points is also .
As an application, we deduce (with an -loss in the exponent) the
bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances
determined by points on an irreducible algebraic curve of constant degree
in the plane that is not a line nor a circle.
\medskip \noindent{\bf (2)} If is two-dimensional, the number of
incidences between and a set of points in is .
\medskip \noindent{\bf (3)} If is three-dimensional and nonlinear, the
number of incidences between and a set of points in is
, provided that no plane contains more than of the points. When , the bound becomes
.
As an application, we prove that the number of incidences between points
and lines in contained in a quadratic hypersurface (which does not
contain a hyperplane) is .
The proofs use, in addition to various tools from algebraic geometry, recent
bounds on the number of incidences between points and algebraic curves in the
plane.Comment: 21 pages, one figur
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
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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