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.
From: Zixuan Zhu [view email]
[v1]
Tue, 3 Mar 2026 20:55:06 UTC (18 KB)
[v2]
Fri, 22 May 2026 08:41:23 UTC (18 KB)
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