  We study various approximation results of solutions of equations $f(x,Y)=0$
where $f(x,Y)\in\mathbb K[[x]][Y]^r$ and $x$ and $Y$ are two sets of variables,
and where some components of the solutions $y(x)\in\mathbb K[[x]]^m$ do not
depend on all the variables $x_j$. These problems have been highlighted by M.
Artin.
