This is a question asking for reference.
I have a proof of the following.
Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions $f_1=f_1(xy,x)$ and $f_2=f_2(xy,y)$ such that $f=f_1+f_2$.
Its proof consists on: 1) deal with the formal problem, 2) deal with the flat terms.
However I do not think such a result is new, it must exist somewhere in the literature. Maybe in some more general context. Does anybody knows a reference for such a result?
Edit: Let me give a bit more detail. The formal series of $f$ can be written as
\begin{equation} \hat f=\sum_{i,j\geq 0}a_{ij}x^iy^j \end{equation}
and can be partitioned as
\begin{equation} \hat f=\hat f_1+\hat f_2=\sum_{i\geq j}a_{ij}(xy)^jx^{i-j}+\sum_{i<j}a_{ij}(xy)^iy^{j-i}. \end{equation}
Then, by Borel's lemma there exist smooth functions $f_1(xy,x)$ and $f_2(xy,y)$ such that $$ f=f_1+f_2+h(x,y) $$
where $h$ is flat. The next step is to "kill" the flat term. I won't go into the detail, but this should be enough to get the idea behind the proof.
