Low-codimensional Subvarieties Inside Dense Multilinear Varieties
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917148596633600 |
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| author | Milićević, Luka |
| author_facet | Milićević, Luka |
| contents | Let $G_1, \dots, G_k$ be finite-dimensional vector spaces over a prime field $\mathbb{F}_p$. Let $V$ be a variety inside $G_1 \times \cdots \times G_k$ defined by a multilinear map. We show that if $|V| \geq c |G_1| \cdots |G_k|$, then $V$ contains a subvariety defined by at most $K(\log_{p} c^{-1} + 1)$ multilinear forms, where $K$ depends on $k$ only. This result is optimal up to multiplicative constant and is relevant to the partition vs. analytic rank problem in additive combinatorics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-codimensional Subvarieties Inside Dense Multilinear Varieties Milićević, Luka Combinatorics Number Theory Let $G_1, \dots, G_k$ be finite-dimensional vector spaces over a prime field $\mathbb{F}_p$. Let $V$ be a variety inside $G_1 \times \cdots \times G_k$ defined by a multilinear map. We show that if $|V| \geq c |G_1| \cdots |G_k|$, then $V$ contains a subvariety defined by at most $K(\log_{p} c^{-1} + 1)$ multilinear forms, where $K$ depends on $k$ only. This result is optimal up to multiplicative constant and is relevant to the partition vs. analytic rank problem in additive combinatorics. |
| title | Low-codimensional Subvarieties Inside Dense Multilinear Varieties |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2512.14529 |