%0 Journal Article
%T Extensions of a result of Elekes and R¨®nyai
%A Ryan Schwartz
%A J¨®zsef Solymosi
%A Frank de Zeeuw
%J Mathematics
%D 2012
%I arXiv
%R 10.1016/j.jcta.2013.06.004
%X 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 $cn^2$ points of an $n\times n\times n$ cartesian product in $\mathbb{R}^3$, then the polynomial has the form $f(x,y)=g(k(x)+l(y))$ or $f(x,y)=g(k(x)l(y))$. They used this to prove a conjecture of Purdy which states that given two lines in $\mathbb{R}^2$ and $n$ points on each line, if the number of distinct distances between pairs of points, one on each line, is at most $cn$, 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 $n\times n\times n\times n$ 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.
%U http://arxiv.org/abs/1206.2717v1