5 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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