# Residue field domination in real closed valued fields

## TL;DR

This paper introduces a notion of residue field domination in real closed valued fields, generalizing stable domination, and characterizes forking behavior in these fields and algebraically closed valued fields.

## Contribution

It defines residue field domination for valued fields and proves domination by residue field sorts over the value group, extending to geometric sorts and laying groundwork for broader theories.

## Key findings

- Real closed valued fields are dominated by residue field sorts over the value group.
- Characterization of forking and h-forking in real closed and algebraically closed valued fields.
- Foundations for extending results to power-bounded T-convex theories.

## Abstract

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field sort and in the geometric sorts. These results characterize forking and \th-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded $T$-convex theory.