Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some homeomorphism? That is, when can I find $\phi:\mathbb{R}^n \to \mathbb{R}^n$ such that $f \circ \phi \in \mathbb{R}[x_1,\ldots, x_n]$? I have tried playing around with the implicit function theorem but haven't gotten far. It feels like I may be missing something very obvious.
Some related questions:
- A necessary condition in the case of $n = 1$ is that $f$ cannot attain the same value infinitely many times (since a polynomial has only finitely many roots). Is this sufficient?
- What if we replace $\mathbb{R}$ by $\mathbb{C}$?
- What if we look at smooth functions instead?
- What about the complex analytic case?