ag.algebraic geometry - Puiseux series for roots of polynomials with smooth coefficients

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 (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 $yto 0$.