Help determine what kind of mapping

Question

If I have that for any equation of the form $$(2^n-1)(1+x+x^2)(1+y+y^2)(1+z+z^2)-2^n(x^2y^2z^2)$$ and given any solution in positive integers $x_0,y_0,z_0$ a map $f(x_0,y_0,z_0) \longrightarrow (2^{n},2^{3n},2^{6n})$, is this just a basic map or for it to be injective what would I need to show ?