Constructions for the Elekes-Szab\'o and Elekes-R\'onyai problems

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–Szab'o problem. We also give a related construction for the Elekes–R'onyai 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 $\mathrm{\mid }P\mathrm{\mid }\gg n^{3\mathrm{/}2}$ on which $f$ takes only $n$ distinct values.