Nemotron-IMO-Bench 2026 A23

Find all sets $A,B\subset\mathbb{R}$ with the following property: if $f: A\times B\to \mathbb{R}$ is a function such that $b\mapsto f(a,b)$ is a polynomial function on $B$ for all $a\in A$ and $a\mapsto f(a,b)$ is a polynomial function on $A$ for all $b\in B$, then $f$ is a polynomial function.