About the algebraic closure of formal power series in several variables

Abstract

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 $K\left[\right[x_1,\dots ,x_r\left]\right]$, $r\ge 2$. More precisely, we view the latter as a subfield of an iterated Puiseux series field $K_r$. On the one hand, given $y_0\in K_r$ 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 $P\in K\left[\right[x_1,\dots ,x_r\left]\right]\left[y\right]$ 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 $y_0$.