Extensions of a result of Elekes and Rónyai

Abstract

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 Rónyai proved that if the graph of a polynomial $f(x,y)$ contains $c{n}^2$ points of an $n\times n\times n$ cartesian product in ${\mathbb{R}}^3$, then the polynomial has one of the forms $f(x,y)=g(k(x)+l(y))$ 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–Rónyai Theorem to a less symmetric cartesian product. This leads to a proof of Purdyʼs conjecture with significantly fewer points on one of the lines. We also extend the Elekes–Rónyai Theorem to $n\times n\times n\times n$ cartesian products, again with an asymmetric version. We finish with a lower bound which shows that our result for asymmetric cartesian products in four dimensions is near-optimal.