I have actually lost the theme of this thread. I even missed - sign on one of your posts.

Here is what I read as the OP:

$f: (\mathbb{Z}\times\mathbb{Z})\to\mathbb{Z}$ defined by the rule $(m,n)\mapsto m^2+n^4$.

If again I have not missed something, it is clearly $f(m,n)\ge0$ so can it be onto?