Approximation and algebraicity in positive characteristic Hahn fields

TL;DR

This paper investigates the algebraic closure properties of certain Hahn fields in positive characteristic, revealing restrictions on supports of elements and implications for decidability in valued field theories.

Contribution

It characterizes the support structure of algebraic elements in positive characteristic Hahn fields and extends results on ramification bounds, impacting decidability questions.

Findings

Supports of elements in the algebraic closure have order type less than ω^ω.

Established bounds on ramification away from p in supports.

Extended Rayner's theorem to residue fields in this context.

Abstract

We study the relative algebraic closure $K$ of $\bar{\mathbb{F}}_p((t))$ inside $\bar{\mathbb{F}}((t^{\mathbb{Q}}))$. We show that the supports of elements in $K$ have order type strictly less than $\omega^\omega$. We also recover a theorem by Rayner giving a bound to the ramification away from $p$ in the support of elements in $K$, and an analogue of Rayner's result for the residue field. This work has applications to the decidability of the first order theory of $\mathbb{F}_p((t^{\mathbb{Q}}))$, and other tame fields, in the language of valued fields with a constant symbol for $t$.