Border subrank via a generalised Hilbert-Mumford criterion
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916684094242816 |
|---|---|
| author | Biaggi, Benjamin Chang, Chia-Yu Draisma, Jan Rupniewski, Filip |
| author_facet | Biaggi, Benjamin Chang, Chia-Yu Draisma, Jan Rupniewski, Filip |
| contents | 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 $Θ(n^{1/(d-1)})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10674 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Border subrank via a generalised Hilbert-Mumford criterion Biaggi, Benjamin Chang, Chia-Yu Draisma, Jan Rupniewski, Filip Algebraic Geometry Computational Complexity 15A69, 14L24, 68Q17 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 $Θ(n^{1/(d-1)})$ 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. |
| title | Border subrank via a generalised Hilbert-Mumford criterion |
| topic | Algebraic Geometry Computational Complexity 15A69, 14L24, 68Q17 |
| url | https://arxiv.org/abs/2402.10674 |