Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And what is the answer if $\mathbb{Q}$ is replaced by $\mathbb{Z}$?

The motivation for this question is Jonas Meyer's comment on the question Polynomial bijection from $\mathbb{Q} \times \mathbb{Q}$ to $\mathbb{Q}$ which says that the explicit determination of an injective polynomial mapping $f: \mathbb{Q}^2 \rightarrow \mathbb{Q}$ is already difficult, and that checking whether the polynomial $x^7+3y^7$ is an example is also.