Take the 2-minute tour ×   $$p(x,y) = x^N + a_{N-1}(y)x^{N-1} +
\ldots + a_0(y), \quad x,y \in \mathbf{C}$$  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 $y\to 0$.  share|improve this
question How would you expand the two roots of
$x^2=exp(-1/y^2)$? –  Piero D'Ancona Aug 13 '10 at 7:10 I would
have thought that you get an asymptotic expansion by repla