Stability of ranks 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.