Suppose that we are given a smooth function $h:\mathbb{R}^n \to \mathbb{R}$ which satisfies $h \circ F= h \circ G$ for two polynomial functions $F,G:\mathbb{R}^m \to \mathbb{R}^n$ (i.e. each component of $F$ and $G$ is a real polynomial of $m$variables). Can we approximate $h$ arbitrarily well by polynomial functions $P_h$ which also satisfy $P_h \circ F= P_h \circ G$ (or at least such that these functions are "close")? How about when $F$ and $G$ are not polynomial but smooth?

1$\begingroup$ In what norm are you approximating? $\endgroup$ – Yemon Choi Dec 8 '11 at 18:52

$\begingroup$ @Yemon: I want to leave that vague for now. If this can be done in some norm I am interested. $\endgroup$ – David Carchedi Dec 8 '11 at 18:56

$\begingroup$ @David: since your domain is noncompact, is $h$ supposed to be bounded? If not, then you will need to have a weight in the norm, which makes life tricky. For instance, what are you hoping for in the case $m=n=1$? $\endgroup$ – Yemon Choi Dec 8 '11 at 19:19

$\begingroup$ @Yemon: Sadly, I have to let $h$ be an arbitrary smooth function, so I cannot assume it's bounded. $\endgroup$ – David Carchedi Dec 8 '11 at 19:25

$\begingroup$ @Yemon: For me though, it's ok not to even have a norm. I just need a reasonably functorial topology on the space of smooth functions for which I can find a convergent sequence of polynomials satisfying the properties I listed. $\endgroup$ – David Carchedi Dec 8 '11 at 19:30
In general not. For $m=n=1$, $F$ and $G$ must be of equal degree (otherwise $P_h\circ F$ anf $P_g\circ F$ are of different degree). But even equal degrees don't guarantee anything: if $F(x)=x$, $G(x) = x+1$, then no polynomial $P$ can fulfill $P\circ F=P\circ G$ (no polynomial is periodic).
You could use the topology of uniform convergence on compact sets. By StoneWeierstrass, any continuous function $h$ can be approximated by a sequence of polynomials $P_n$ in this topology, and then $P_n \circ F  P_n \circ G$ approximates $h \circ F  h \circ G = 0$ in this topology (because if $K \subset {\mathbb R}^m$ is compact, $F(K) \cup G(K)$ is compact).

$\begingroup$ The statement "if $P_n$ approximates $h$, then $P_n \circ FP_n \circ G$ approaches $0$" is true whenever composition and subtraction are continuous maps in whatever topology on smooth functions you're considering. Since most (all?) reasonable topologies will satisfy that property, in general this will not give an extra constraint. $\endgroup$ – Will Sawin Dec 9 '11 at 6:42