Border subrank via a generalised Hilbert-Mumford criterion.(English)Zbl 07967031
Summary: 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 \(\operatorname{\Theta}( n^{1 / ( d - 1 )})\) 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.
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