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.

Added on Aug 8, 2013: SJR's nice answer still leaves the following 3 problems open:

1. Is there at all an injective polynomial mapping from $\mathbb{Q}^2$ to $\mathbb{Q}$?

2. Would a positive answer to Hilbert's Tenth Problem over $\mathbb{Q}$ imply that surjectivity of polynomial functions $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is algorithmically decidable?

3. Hilbert's Tenth Problem over $\mathbb{Q}$.