Valued fields with a total residue map

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 $0$.

arXiv comment13 pages; streamlined some parts and improved the presentation