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 $\mathbb{Q}$][1]
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. 

<b>Added on Aug 8, 2013:</b> 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 $\mathbb{Q}$.


  [1]: http://mathoverflow.net/questions/21003/