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.

翻译：我们证明，对于$(\mathbb{C}^n)^{\otimes d}$中足够一般的张量，当$n \to \infty$时，其边界子秩为$\mathcal{O}(n^{1/(d-1)})$。由于该结果与Derksen-Makam-Zuiddam近期建立的通用（非边界）子秩增长率$\Theta(n^{1/(d-1)})$相匹配，我们发现通用边界子秩具有相同的增长率。在证明过程中，我们使用了Hilbert-Mumford准则的广义形式，相信该推广方法本身也具有独立的研究价值。