AI 中文总结
研究有限秩超简单理论中稳定分叉猜想的主要情形，通过将佩雷茨在秩3的工作转化为存在性定理，并结合多实验参数可定义性论证，证明决定分叉稳定性的拟阵集可有限，为该猜想提供新证明。
AI 中文摘要
我们证明了稳定分叉猜想的主要情形之一，即在有限秩超简单理论中基于一个基的分叉关系的稳定性，在每个秩中由有限多个预几何结构决定。稳定分叉猜想的这种情形长期以来有一种隐含的、广为人知的预几何解释：存在一组拟阵$\mathcal{G}_{n}$，使得秩$n$中的分叉不稳定性等同于某个秩为一的部分类型（在有限集上）的预几何结构嵌入$\mathcal{G}_{n}$中的一个拟阵。我们的贡献在于表明，确定秩$n$中基于分叉稳定性的这组拟阵$\mathcal{G}_{n}$可以选择为有限的。我们证明的主要部分已由佩雷茨在秩为3时完成，但不能按所述扩展到更高秩（并且可能在较弱意义上直接扩展到更高秩，通过缩小术语）。然而，我们在秩$n>3$时获得了佩雷茨工作的充分替代：我们将佩雷茨原来的通用结果转化为一个存在性定理。我们证明的其余部分改进了来自多实验参数可定义性的一个论证，该论证源于李、梅什卡特、奥夫钦尼科夫、皮莱、波古丁和斯坎伦在应用模型理论中的工作。
英文摘要
We show that one of the main cases of the stable forking conjecture, stability of the forking relation over a base in a finite-rank supersimple theory, is determined by finitely many pregeometries in each rank. This case of the stable forking conjecture has long had an implicitly well-known pregeometric interpretation: there is a set of matroids $\mathcal{G}_{n}$ such that the forking instability in rank $n$ is equivalent to the pregeometry on some rank-one partial type (over a finite set) embedding a matroid in $\mathcal{G}_{n}$. Our contribution is to show that this set of matroids $\mathcal{G}_{n}$, determining based forking stability in rank $n$, can be chosen to be finite. The main part of our proof was already accomplished in rank $3$ by Peretz, but does not extend as stated to higher ranks (and may or may not directly extend in a weaker sense to higher ranks, by shrinking terms). However, we obtain a sufficient substitute for Peretz's work in ranks $n > 3$: we turn Peretz's original universal result into an existence theorem. The rest of our proof refines an argument from multi-experiment parameter definability, originating from work in applied model theory by Li, Meshkat, Ovchinnikov, Pillay, Pogudin and Scanlon.
Comments34 pages
