Valued fields with a total residue map

TL;DR

This paper investigates valued fields with a total residue map, showing the theory lacks a model companion and that certain fields with this structure are undecidable, highlighting limits of definability and decidability in valued field theories.

Contribution

It introduces and analyzes the theory of valued fields with a total residue map, proving non-existence of a model companion and undecidability results for these structures.

Findings

The theory $ ext{VF}_{ ext{res}, ext{iota}}$ does not admit a model companion.

The power series field $(k( ext!(t) ), ext{res})$ is undecidable for infinite $k$.

The complex Laurent series field with residue map is undecidable.

Abstract

When $k$ is a finite field, Becker-Denef-Lipschitz (1979) observed that the total residue map $\text{res}:k(\!(t)\!)\to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for $t$. Driven by this observation, we study the theory $\text{VF}_{\text{res},\iota}$ of valued fields equipped with a linear form $\text{res}:K\to k$ which specializes to the residue map on the valuation ring. We prove that $\text{VF}_{\text{res},\iota}$ does not admit a model companion. In addition, we show that the power series field $(k(\!(t)\!),\text{res})$, equipped with such a total residue map, is undecidable whenever $k$ is an infinite field. As a consequence, we get that $(\mathbb{C}(\!(t)\!), \text{Res}_0)$ is undecidable, where $\text{Res}_0:\mathbb{C}(\!(t)\!)\to \mathbb{C}:f\mapsto \text{Res}_0(f)$ maps $f$ to its complex residue at $0$.