Elimination of imaginaries in $\mathbb{C}((\Gamma))$

TL;DR

This paper investigates elimination of imaginaries in certain henselian valued fields with algebraically closed residue fields, focusing on the complexity of the value group and establishing conditions for weak or full elimination.

Contribution

It provides new elimination results for valued fields with finite spines and dp-minimal value groups, introducing quotient sorts and codes for definable submodules.

Findings

Finite spines valued fields admit weak elimination of imaginaries with added sorts.

Dp-minimal valued fields eliminate imaginaries with additional quotient sorts and constants.

Results depend on the complexity of the value group and definable convex subgroups.

Abstract

In this paper we study elimination of imaginaries in some classes of henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first in the case where the ordered abelian group has finite spines, and then prove a better result for the dp-minimal case. An ordered abelian with finite spines weakly eliminates imaginaries once we add sorts for the quotient groups $\Gamma/ \Delta$ for each definable convex subgroup $\Delta$, and sorts for the quotient groups $\Gamma/ \Delta+ l\Gamma$ where $\Delta$ is a definable convex subgroup and $l \in \mathbb{N}_{\geq 2}$. We refer to these sorts as the quotient sorts. We prove the following two theorems: Theorem: Let $K$ be a valued field of equicharacteristic zero, residue field algebraically closed and value group with finite spines. Then $K$ admits weak elimination of imaginaries once we add codes for all the definable $\mathcal{O}$-submodules of $K^{n}$ for each $n \in \mathbb{N}$, and the quotient sorts for the value group. Theorem: Let $K$ be a henselian valued field of equicharacteristic zero, residue field algebraically closed and dp-minimal value group. Then $K$ eliminates imaginaries once we add codes for all the definable $\mathcal{O}$-submodules of $K^{n}$ for each $n \in \mathbb{N}$, the quotient sorts for the value group and constants to distinguish the elements of the finite groups $\Gamma/n\Gamma$ for each $ n \in \mathbb{N}_{\geq2}$.