83 research outputs found
Extensions of a result of Elekes and R\'onyai
Many problems in combinatorial geometry can be formulated in terms of curves
or surfaces containing many points of a cartesian product. In 2000, Elekes and
R\'onyai proved that if the graph of a polynomial contains points of an
cartesian product in , then the polynomial
has the form or . They used this to
prove a conjecture of Purdy which states that given two lines in
and points on each line, if the number of distinct distances between pairs
of points, one on each line, is at most , then the lines are parallel or
orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian
product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on
an cartesian product and an asymmetric cartesian
product. We give a proof of a variation of Purdy's conjecture with fewer points
on one of the lines. We finish with a lower bound for our main result in one
dimension higher with asymmetric cartesian product, showing that it is
near-optimal.Comment: 23 page
Crossing Patterns of Segments
AbstractIt is shown that for every c>0 there exists c′>0 satisfying the following condition. Let S be a system of n straight-line segments in the plane, which determine at least cn2 crossings. Then there are two disjoint at least c′n-element subsystems, S1, S2⊂S, such that every element of S1 crosses all elements of S2
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