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On the number of ordinary conics
We prove a lower bound on the number of ordinary conics determined by a
finite point set in . An ordinary conic for a subset of
is a conic that is determined by five points of , and
contains no other points of . Wiseman and Wilson proved the
Sylvester-Gallai-type statement that if a finite point set is not contained in
a conic, then it determines at least one ordinary conic. We give a simpler
proof of their result and then combine it with a result of Green and Tao to
prove our main result: If is not contained in a conic and has at most
points on a line, then determines ordinary conics.
We also give a construction, based on the group structure of elliptic curves,
that shows that the exponent in our bound is best possible
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