Rank and Independence of Imaginaries in Proper Pairs of ACF
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916037583175680 |
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| author | Zhu, Zixuan |
| author_facet | Zhu, Zixuan |
| contents | 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 $ω*\mathbb N + \mathbb Z$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03518 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rank and Independence of Imaginaries in Proper Pairs of ACF Zhu, Zixuan Logic 03C45, 03C60 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 $ω*\mathbb N + \mathbb Z$ 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. |
| title | Rank and Independence of Imaginaries in Proper Pairs of ACF |
| topic | Logic 03C45, 03C60 |
| url | https://arxiv.org/abs/2603.03518 |