域扩张下秩的稳定性
Stability of ranks under field extensions
AI总结：
本文研究了张量秩在域扩张下的稳定性，证明了分析秩的稳定性，并解决了多个相关猜想。
摘要：
本文研究了张量秩在域扩张下的稳定性。我们的主要贡献有四点：（1）我们证明了分析秩在域扩张下是稳定的。（2）我们建立了分区秩与分析秩猜想与分区秩稳定性猜想之间的等价性，并证明它们等价于其他两个重要猜想。（3）我们解决了Adiprasito-Kazhdan-Ziegler关于线性子空间切片秩在域扩张下稳定性的猜想。（4）作为（1）的应用，我们证明了几何秩与分析秩在常数因子范围内相等。
英文摘要：
This paper studies the stability of tensor ranks under field extensions. Our main contributions are fourfold: (1) We prove that the analytic rank is stable under field extensions. (2) We establish the equivalence between the partition rank vs. analytic rank conjecture and the stability conjecture for partition rank. We also prove that they are equivalent to other two important conjectures. (3) We resolve the Adiprasito-Kazhdan-Ziegler conjecture on the stability of the slice rank of linear subspaces under field extensions. (4) As an application of (1), we show that the geometric rank is equal to the analytic rank up to a constant factor.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
