Algebraic series and valuation rings over nonclosed fields

TL;DR

This paper provides a simple criterion for when a formal power series over an algebraic closure is algebraic over a field of formal Laurent series, and characterizes certain valuation rings over two-dimensional local domains.

Contribution

It introduces a straightforward necessary and sufficient condition for algebraicity of power series over nonclosed fields and characterizes valuation rings with specific properties over two-dimensional domains.

Findings

A simple criterion for algebraicity of power series over nonclosed fields.

Characterization of valuation rings dominating two-dimensional local domains.

Insight into rank behavior after completion of birational extensions.

Abstract

Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $\sigma$ is a formal power series in $x_1,...,x_n$ with coefficints in the algebraic closure of $k$. We give a very simple necessary and sufficient condition for $\sigma$ to be algebraic over $k((x_1,...,x_n))$. As an application of our methods, we give a characterization of valuation rings $V$ which dominate an excellent, Noetherian local domain $R$ of dimension two, and such that the rank increases after passing to the completion of a birational extension of $R$.