About the algebraic closure of formal power series in several variables

TL;DR

This paper investigates the algebraic closure of formal power series fields in multiple variables over characteristic zero fields, providing algorithms for polynomial annihilators and explicit formulas for roots in terms of coefficients.

Contribution

It introduces an algorithm to reconstruct annihilating polynomials for algebraic elements and derives explicit formulas for roots of polynomials with simple roots in multivariate power series fields.

Findings

Algorithm for reconstructing annihilating polynomials

Closed form formulas for roots in terms of coefficients

Enhanced understanding of algebraic closure in multivariate power series fields

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 $K[[x_1,\ldots,x_r]]$, $r\geq 2$. 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 $y_0$.