Rank and Independence of Imaginaries in Proper Pairs of ACF

TL;DR

This paper introduces a new geometric rank for imaginaries in the theory of beautiful pairs of algebraically closed fields, refining existing ranks and providing explicit forking independence criteria.

Contribution

It defines a geometric rank on imaginaries in $T_P$, extending SU-rank and enabling explicit forking independence characterization.

Findings

Geometric rank coincides with SU-rank on real tuples.

Provides an explicit criterion for forking independence in $T_P^{ ext{eq}}$.

Refines the understanding of imaginaries in algebraically closed fields.

Abstract

Let $T_P$ be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of $T_P$, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in $T_P$, we define an additive rank on imaginaries of $T_P$, called the geometric rank. It takes values in $\omega*\mathbb N + \mathbb Z$ and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in $T_P^{\mathrm{eq}}$, from which we derive an explicit criterion for determining forking independence.