Border apolarity and varieties of sums of powers
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913234235162624 |
|---|---|
| author | Mańdziuk, Tomasz Ventura, Emanuele |
| author_facet | Mańdziuk, Tomasz Ventura, Emanuele |
| contents | We study border varieties of sums of powers ($\underline{\mathrm{VSP}}$'s for short), recently introduced by Buczyńska and Buczyński, parameterizing border rank decompositions of a point (e.g. of a tensor or a homogeneous polynomial) with respect to a smooth projective toric variety and living in the Haiman-Sturmfels multigraded Hilbert scheme. Their importance stems from the role of border tensor rank in theoretical computer science, especially in the estimation of the exponent of matrix multiplication, a fundamental and still unknown quantity in the theory of computation. We compare $\underline{\mathrm{VSP}}$'s to other well-known loci in the Hilbert scheme, parameterizing scheme-theoretic versions of decompositions. The latter ones are crucial in that they naturally explain the existing severe barriers to giving good lower bounds on ranks. We introduce the notion of border identifiability and provide sufficient criteria for its appearance, which rely on the multigraded regularity of Maclagan and Smith. We link border identifiability to wildness of points. Finally, we determine $\underline{\mathrm{VSP}}$'s in several instances and regimes, in the contexts of tensors and homogeneous polynomials. These include concise $3$-tensors of minimal border rank and in particular of border rank three. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19625 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Border apolarity and varieties of sums of powers Mańdziuk, Tomasz Ventura, Emanuele Algebraic Geometry Primary 14C05, Secondary 14M25, 15A69, 68Q17 We study border varieties of sums of powers ($\underline{\mathrm{VSP}}$'s for short), recently introduced by Buczyńska and Buczyński, parameterizing border rank decompositions of a point (e.g. of a tensor or a homogeneous polynomial) with respect to a smooth projective toric variety and living in the Haiman-Sturmfels multigraded Hilbert scheme. Their importance stems from the role of border tensor rank in theoretical computer science, especially in the estimation of the exponent of matrix multiplication, a fundamental and still unknown quantity in the theory of computation. We compare $\underline{\mathrm{VSP}}$'s to other well-known loci in the Hilbert scheme, parameterizing scheme-theoretic versions of decompositions. The latter ones are crucial in that they naturally explain the existing severe barriers to giving good lower bounds on ranks. We introduce the notion of border identifiability and provide sufficient criteria for its appearance, which rely on the multigraded regularity of Maclagan and Smith. We link border identifiability to wildness of points. Finally, we determine $\underline{\mathrm{VSP}}$'s in several instances and regimes, in the contexts of tensors and homogeneous polynomials. These include concise $3$-tensors of minimal border rank and in particular of border rank three. |
| title | Border apolarity and varieties of sums of powers |
| topic | Algebraic Geometry Primary 14C05, Secondary 14M25, 15A69, 68Q17 |
| url | https://arxiv.org/abs/2310.19625 |