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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.12319 |
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| _version_ | 1866914016120537088 |
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| author | Blatter, Andreas Draisma, Jan Rupniewski, Filip |
| author_facet | Blatter, Andreas Draisma, Jan Rupniewski, Filip |
| contents | Restriction is a natural quasi-order on $d$-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field -- namely, that it is a well-quasi-order: it admits no infinite antichains and no infinite strictly decreasing sequences. This result, reminiscent of the graph minor theorem, has important consequences for an arbitrary restriction-closed tensor property $X$. For instance, $X$ admits a characterisation by finitely many forbidden restrictions and can be tested by looking at subtensors of a fixed size. Our proof involves an induction over polynomial generic representations, establishes a generalisation of the tensor restriction theorem to other such representations (e.g. homogeneous polynomials of a fixed degree), and also describes the coarse structure of any restriction-closed property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_12319 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Tensor Restriction Theorem over Finite Fields Blatter, Andreas Draisma, Jan Rupniewski, Filip Algebraic Geometry Restriction is a natural quasi-order on $d$-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field -- namely, that it is a well-quasi-order: it admits no infinite antichains and no infinite strictly decreasing sequences. This result, reminiscent of the graph minor theorem, has important consequences for an arbitrary restriction-closed tensor property $X$. For instance, $X$ admits a characterisation by finitely many forbidden restrictions and can be tested by looking at subtensors of a fixed size. Our proof involves an induction over polynomial generic representations, establishes a generalisation of the tensor restriction theorem to other such representations (e.g. homogeneous polynomials of a fixed degree), and also describes the coarse structure of any restriction-closed property. |
| title | A Tensor Restriction Theorem over Finite Fields |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2211.12319 |