Constructions for the Elekes-Szabó and Elekes-Rónyai problems

Abstract

We give a construction of a non-degenerate polynomial $F\in \mathbb{R}[x,y,z]$ and a set $A$ of cardinality $n$ such that $\left|Z(F)\cap (A\times A\times A)\right|\gg n^{\frac{3}{2}}$, thus providing a new lower bound construction for the Elekes--Szabó problem. We also give a related construction for the Elekes--Rónyai problem restricted to a subgraph. This consists of a polynomial $f\in \mathbb{R}[x,y]$ that is not additive or multiplicative, a set $A$ of size $n$, and a subset $P\subset A\times A$ of size $|P|\gg n^{3/2}$ on which $f$ takes only $n$ distinct values.