Background:
I had a thought on algebraically describing the set of all smooth (in the sense that it has a derivative of any order, $C^\infty$) functions from an euclidean space $\mathbb{R}^n$ to $\mathbb{R}$. Call this set $M_n$ for the dimension $n$. I know it's an algebra, but it obviously has a much stricter structure. For example, if $f:\mathbb{R}^m\rightarrow\mathbb{R}$ is a smooth function and $g_1,\dots,g_m\in M_n$ then $f(g_1,\dots,g_m)\in M_n$ too. Now my question is, can I bound the value of $m$?
A concrete example question:
Is there a finite subset $G$ of $M_3$ such that the subset $P$ of $M_3$, constructed as the minimal set which satisfies:
- $G\subseteq P$,
- $(\forall f\in M_2)(\forall g,h\in P)\;f(g,h)\in P$,
is equal to $M_3$ ($P=M_3$)?
My idea:
I am imagining that if I tile $\mathbb{R}^3$ with cubes (make a grid) and then "randomly" choose some subset of cubes and stuff these cubes with bump functions, while leaving other cubes map to zero, then I can prove that this function is not in $P$. However, I can't realize this idea.