22 research outputs found
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
On the number of classes of triangles determined by points in
Let be a set of points in the Euclidean plane, where a positive
proportion of points lies off a single straight line. This note points out two
facts concerning the number of equivalence classes of triangles that
determines, namely that (i) determines different equivalence
classes of congruent triangles, and (ii) determines different equivalence classes of similar triangles. The first fact follows
from the recent theorem by Guth-Katz on point-line incidences in . The
second one, perhaps not so well known, is due to Solymosi and Tardos.Comment: 6p
An improved bound on the number of point-surface incidences in three dimensions
We show that points and smooth algebraic surfaces of bounded degree
in satisfying suitable nondegeneracy conditions can have at most
incidences, provided that any
collection of points have at most O(1) surfaces passing through all of
them, for some . In the case where the surfaces are spheres and no
three spheres meet in a common circle, this implies there are point-sphere incidences. This is a slight improvement over the previous
bound of for an (explicit) very
slowly growing function. We obtain this bound by using the discrete polynomial
ham sandwich theorem to cut into open cells adapted to the set
of points, and within each cell of the decomposition we apply a Turan-type
theorem to obtain crude control on the number of point-surface incidences. We
then perform a second polynomial ham sandwich decomposition on the irreducible
components of the variety defined by the first decomposition. As an
application, we obtain a new bound on the maximum number of unit distances
amongst points in .Comment: 17 pages, revised based on referee comment
On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves
A long standing conjecture of Richter and Thomassen states that the total
number of intersection points between any simple closed Jordan curves in
the plane, so that any pair of them intersect and no three curves pass through
the same point, is at least .
We confirm the above conjecture in several important cases, including the
case (1) when all curves are convex, and (2) when the family of curves can be
partitioned into two equal classes such that each curve from the first class is
touching every curve from the second class. (Two curves are said to be touching
if they have precisely one point in common, at which they do not properly
cross.)
An important ingredient of our proofs is the following statement: Let be
a family of the graphs of continuous real functions defined on
, no three of which pass through the same point. If there are
pairs of touching curves in , then the number of crossing points is
.Comment: To appear in SODA 201
Unit Distances in Three Dimensions
We show that the number of unit distances determined by n points in R^3 is
O(n^{3/2}), slightly improving the bound of Clarkson et al. established in
1990. The new proof uses the recently introduced polynomial partitioning
technique of Guth and Katz [arXiv:1011.4105]. While this paper was still in a
draft stage, a similar proof of our main result was posted to the arXiv by
Joshua Zahl [arXiv:1104.4987].Comment: 13 page