If 


$$p(x,y) = x^N + a\_{N-1}(y)x^{N-1} + \ldots + a\_0(y), \quad x,y \in \mathbf{C}$$


is a monic polynomial in $x$, and the coefficients $a\_j$ are analytic functions of $y$, then the roots of $p$ have expansions in [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_series) (in powers of $y^{1/m}$ for some $m$) which are convergent for $y$ sufficiently close to 0.


Is this true in an asymptotic sense when the $a\_j$ are only assumed to be smooth? i.e. Do the roots have asymptotic expansions which are formal Puiseux series (not necessarily convergent)?


For $A$ to have an asymptotic expansion $A \sim B\_1 + B\_2 + \ldots$ with respect to some grading $\mathcal{O}(n)$ means that $B\_n \in \mathcal{O}(n)$, and for each $N$, $A - \sum\_{n=1}^N B\_n \in O(N+1)$. Here $\mathcal{O}(n)$ means $\mathcal{O}(|y|^{n/m})$ in the usual big-O notation, as $y\to 0$. 



Puiseux series for roots of polynomials with smooth coefficients
The constructive proof of the Newton-Puiseux theorem works *formally*, and a posteriori one can show that if the original coefficients are convergent, then the Puiseux series are convergent (see the details in Casas-Alvero's book, "Singularities of Plane Curves"). So Torsten Ekedahl's comment is right, of course.
However, I wonder how useful this would be when the power series don't converge to the original smooth functions.


