Abstract:

Many metric combinatorial problems (among others those on the
distribution of distances defined by finite point sets of the
Euclidean plane) turned out to be extremely difficult and
have been unsolved (and almost hopeless) for decades.
One possible reason for this phenomenon may be that such questions,
if attacked by algebraic methods, lead to complicated polynomials.
(E.g., expressing the relation of the three distances from point X
to arbitrary points A,B,C of the plane.)
We present a (partly algebraic) technique which provides new progress
in several metric questions (also on slopes and collinear triplets).
The essence of the method is an interaction between Polynomial Algebra
and Szemeredi-Trotter type results of Combinatorial Geometry
(flavored with some elementary Calculus).

(Includes joint work with L.Ronyai, E.Szabo and Z.Kiraly)

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