Elimination results for tame fields with finite residue fields

TL;DR

This paper investigates the model theory of Hahn series fields over finite fields with the $t$-adic valuation, establishing quantifier elimination results in the language of valued fields.

Contribution

It proves that every formula in this setting is equivalent to an existential formula involving polynomial equations, extending previous work on tame fields.

Findings

Every formula is equivalent to an existential polynomial equation.

The results build on and extend work by Kuhlmann and Lisinski.

The study focuses on the Hahn series field over finite fields with the $t$-adic valuation.

Abstract

Building on work of Kuhlmann and Lisinski, we study the theory of the Hahn series field $\mathbb{F}_{q}(\!(\mathbb{Q})\!)$, over a finite field $\mathbb{F}_{q}$, equipped with the $t$-adic valuation, in a language of valued fields. We prove that every formula is equivalent to a formula $\exists y\colon f(x_{1},\ldots,x_{n},y)=0$, for a polynomial $f\in\mathbb{Z}[x_{1},\ldots,x_{n},y]$.