Border subrank via a generalised Hilbert-Mumford criterion

Abstract

We show that the border subrank of a sufficiently general tensor in ${\left(C^n\right)}^{\otimes d}$ is $O\left(n^{1/(d-1)}\right)$ for $n\to \infty$. Since this matches the growth rate $\Theta \left(n^{1/(d-1)}\right)$ for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest.