About the algebraic closure of the field of power series in several variables in characteristic zero

TL;DR

This paper constructs algebraically closed fields containing power series fields in several variables over characteristic zero, using Abhyankar valuations and Newton-Puiseux methods, and extends classical theorems to new valuation cases.

Contribution

It introduces a method to build algebraically closed fields from power series fields in multiple variables using valuation theory and generalizes the Abhyankar-Jung Theorem for specific polynomial cases.

Findings

Constructed algebraically closed fields via Abhyankar valuations.

Extended the Abhyankar-Jung Theorem to weighted homogeneous discriminants.

Analyzed monomial valuations in the context of power series fields.

Abstract

We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and are constructed via the Newton-Puiseux method. Then we study more carefully the case of monomial valuations and we give a result generalizing the Abhyankar-Jung Theorem for monic polynomials whose discriminant is weighted homogeneous.