285 research outputs found
On the use of Klein quadric for geometric incidence problems in two dimensions
We discuss a unified approach to a class of geometric combinatorics incidence
problems in , of the Erd\"os distance type. The goal is obtaining the
second moment estimate, that is given a finite point set and a function
on , an upper bound on the number of solutions of E.g., is the Euclidean distance in the plane, sphere, or a
sheet of the two-sheeted hyperboloid.
Our tool is the Guth-Katz incidence theorem for lines in , but
we focus on how the original problem is made amenable to it. This
procedure was initiated by Elekes and Sharir, based on symmetry considerations.
However, symmetry considerations can be bypassed or made implicit. The
classical Pl\"ucker-Klein formalism for line geometry enables one to directly
interpret a solution of as intersection of two lines in .
This allows for a very brief argument extending the Euclidean plane distance
argument to the spherical and hyperbolic distances. We also find instances of
the question without underlying symmetry group.
The space of lines in the three-space, the Klein quadric , is
four-dimensional. We start out with an injective map , from a pair of points in to a line in and seek a
combinatorial problem in the form , which can be solved by applying the
Guth-Katz theorem to the set of lines in question. We identify a few new such
problems and generalise the existing ones.Comment: Theorem 5', implicit in the earlier verisons has been stated
explicitly in this ArXiv version, giving a family of applications of the
Guth-Katz theorem to sum-product type quantities, with no underlying symmetry
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Partial ovoids and partial spreads in finite classical polar spaces
We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces
Partial Ovoids and Partial Spreads of Classical Finite Polar Spaces
2000 Mathematics Subject Classification: 05B25, 51E20.We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces.The research of the fourth author was also supported by the Project Combined algorithmic and the oretical study of combinatorial structur es between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Science
Point-plane incidences and some applications in positive characteristic
The point-plane incidence theorem states that the number of incidences
between points and planes in the projective three-space over a
field , is where is the maximum number
of collinear points, with the extra condition if has
characteristic . This theorem also underlies a state-of-the-art
Szemer\'edi-Trotter type bound for point-line incidences in , due to
Stevens and de Zeeuw.
This review focuses on some recent, as well as new, applications of these
bounds that lead to progress in several open geometric questions in , for
. These are the problem of the minimum number of distinct nonzero
values of a non-degenerate bilinear form on a point set in , the analogue
of the Erd\H os distinct distance problem in and additive energy
estimates for sets, supported on a paraboloid and sphere in . It avoids
discussing sum-product type problems (corresponding to the special case of
incidences with Cartesian products), which have lately received more attention.Comment: A survey, with some new results, for the forthcoming Workshop on
Pseudorandomness and Finite Fields in at RICAM in Linz 15-19 October, 2018;
24p
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