Border subrank via a generalised Hilbert-Mumford criterion

Abstract

We show that the border subrank of a sufficiently general tensor in $({\mathbb{C}}^n{)}^{\otimes d}$ is $\mathcal{O}(n^{1/(d-1)})$ for $n\to \infty$. Since this matches the growth rate $\Theta (n^{1/(d-1)})$ 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.

14 pages, 2 figures, final version