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 $F$ vanishes on $\Omega(n^{3/2})$ points of $A \times A \times A$, 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 $\Omega(n^{3/2})$ on which $f$ takes only $n$ distinct values.