Implicit function whose domain is $\mathbb R^2$

Question

Let $F(x, y, z)=z^3+3z+2x^4+y^2-x^2-2y$. I want to show that the equation $F(x, y, z)=0$ defines a $C^2$ function $z=f(x, y)$ whose domain is $\mathbb R^2$.

By the implicit function theorem, it's easy to see that such an $f$ exists in a neighborhood of $(0, 0)$, however, I don't know how to show that we can extend this function to $\mathbb R^2$.

Answer 1

Well, I got it. The function $t \rightarrow t^3+3t$ is bijective, therefore given $x, y$, there exists only one $z$ such that $z^3+3z=-2x^4-y^2+x^2+2y$, so only one such $f$ exists. The implicit function theorem is used to show that this $f$ is $C^\infty$.