  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 $|P|\gg n^{3/2}$ on
which $f$ takes only $n$ distinct values.
