Uniformity for limits of tensors
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913237818146816 |
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| author | Bik, Arthur Draisma, Jan Eggermont, Rob Snowden, Andrew |
| author_facet | Bik, Arthur Draisma, Jan Eggermont, Rob Snowden, Andrew |
| contents | There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the locus with "border rank" $\le r$) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank $\le r$ do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on $\mathbf{GL}$-varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on $\mathbf{GL}$-varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19866 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniformity for limits of tensors Bik, Arthur Draisma, Jan Eggermont, Rob Snowden, Andrew Algebraic Geometry Representation Theory There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the locus with "border rank" $\le r$) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank $\le r$ do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on $\mathbf{GL}$-varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on $\mathbf{GL}$-varieties. |
| title | Uniformity for limits of tensors |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2305.19866 |