INCIDENCES AND COINCIDENCES

by Gyorgy Elekes

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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