Stability of ranks under field extensions

TL;DR

This paper investigates how tensor ranks behave under field extensions, proving stability for certain ranks, establishing key conjecture equivalences, and resolving a notable conjecture on slice rank stability.

Contribution

It proves the stability of the analytic rank, establishes equivalences among major conjectures, and resolves the Adiprasito-Kazhdan-Ziegler conjecture on slice rank stability.

Findings

Analytic rank is stable under field extensions.

Partition rank conjecture is equivalent to the analytic rank conjecture.

Slice rank of linear subspaces remains stable under field extensions.

Abstract

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.