  Let $K$ be a field of characteristic zero. We deal with the algebraic closure
of the field of fractions of the ring of formal power series
$K[[x_1,\ldots,x_r]]$, $r\geq 2$. More precisely, we view the latter as a
subfield of an iterated Puiseux series field $\mathcal{K}_r$. On the one hand,
given $y_0\in \mathcal{K}_r$ which is algebraic, we provide an algorithm that
reconstructs the space of all polynomials which annihilates $y_0$ up to a
certain order (arbitrarily high). On the other hand, given a polynomial $P\in
K[[x_1,\ldots,x_r]][y]$ with simple roots, we derive a closed form formula for
the coefficients of a root $y_0$ in terms of the coefficients of $P$ and a
fixed initial part of $y_0$.
