On The Number Of Ordinary Conics

We prove a lower bound on the number of ordinary conics determined by a finite point set in R-2. An ordinary conic for S subset of R-2 is a conic that is determined by five points of S and contains no other points of S. 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 this statement and then combine it with a theorem of Green and Tao to prove our main result: If S is not contained in a conic and has at most c vertical bar S vertical bar points on a line, then S determines Omega(c)(vertical bar 5 vertical bar(4)) ordinary conics. We also give constructions, based on the group law on elliptic curves, that show that the exponent in our bound is best possible.


Published in:
Siam Journal On Discrete Mathematics, 30, 3, 1644-1659
Year:
2016
Publisher:
Philadelphia, Siam Publications
ISSN:
0895-4801
Keywords:
Laboratories:




 Record created 2016-11-21, last modified 2018-03-17


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