Eric, now that I'm thinking about your construction, why do you get a finite disjoint union of open cells?

Thanks Eric for the instructive exemple. I can now better appreciate the constraints that come with the definition of a (finite) CW complex where the gluing does not affect the open cells but only their boundaries.

Dear Alexander, I think that what was on my mind when I wrote this post was given $n\in\mathbf{Z}_{\geq 3}$ and $a,b$ algebraic real numbers, can you give a criterion which says when is $z^{n}-(a+bi)$ positive solvable.

Dear Alexander, already in the answer that I had posted on Sep 21, 2012, I was quoting Brian Conrad's note: "radical tower and roots of unity"

So I understood well what is going on. The sketch of the proof in (b) is valid. In fact, the result is completely general. If one deals with a linear system of homogeneous ODE's in $g$ variables of order $n_1,n_2,\ldots,n_g$ (where ONLY ONE variable appears in each equation) then the solution space will be a vector space of dimension $n_1n_2\ldots n_g$ over the field of meromorphic functions in one variable. Of course this a very special kind of PDEs system.