Let $k$ be a positive integer greater than $1$ and suppose that $F \in \mathbb{Z}[x_{1}, \ldots, x_{k}]$.

Can we always find a natural number $n(k)$ and $f_{1}, \ldots f_{n(k)} \in \mathbb{Z}[x]$ such that

$\displaystyle F(\bigoplus_{j=1}^{k} \mathbb{Z}) = \bigcup_{j=1}^{n(k)} f_{j}(\mathbb{Z})$ ?

Think this question is really cool. What do you guys think?