Zixuan Zhu : Rank and Independence of Imaginaries in Proper Pairs of ACF

Posted by James Freitag , part of the Logic Seminar.

At

March 17, 2026, 3 p.m.

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636 SEO

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 all imaginaries, called the geometric rank. It takes values in ω*N+Z and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in T_P^eq. As a consequence, we derive an explicit criterion for determining forking independence.