Triple lines on curves

Frank de Zeeuw
École Polytechnique Fédérale de Lausanne (EPFL)

I will discuss a result of Elekes and Szabo, which basically says that any n points on an algebraic curve in R^2 determine O(n^{2-eps}) triple lines, unless the curve has degree three or less. It is a consequence of a more general theorem of Elekes and Szabo, which bounds the intersection of a surface in R^3 with an nxnxn Cartesian product. In (IPAM-based) work with Orit Raz and Micha Sharir, we improved this general theorem, thereby improving the bound in the result on triple lines to O(n^{11/6}).

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