切片秩方法在加性组合学中的局限性
Limitations of the slice rank method in additive combinatorics
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中文总结 AI 辅助
本文证明对于k≥4，仅在k项等差数列或k向日葵关系上非零的张量在任何域上具有最大切片秩，从而排除了切片秩方法在这些情形中的指数节省。
中文摘要 AI 辅助
切片秩方法为奇特征有限向量空间中无三项等差数列的集合以及固定底集子集的三向日葵自由族提供了指数界。我们证明，对于$k\ge4$，每个仅在$k$项等差数列关系或$k$向日葵关系上非零的张量，在任何系数域上都具有最大切片秩。当支撑仅在两两不同的输入上被规定时，我们获得可比较的下界，这同样排除了指数节省。
英文摘要
The slice rank method gives exponential bounds for sets with no three-term arithmetic progression in finite vector spaces of odd characteristic and for three-sunflower-free families of subsets of a fixed ground set. We show that for $k\ge4$, every tensor that is nonzero exactly on the $k$-term arithmetic progression relation or the $k$-sunflower relation has maximal slice rank over every coefficient field. When the support is prescribed only on pairwise distinct inputs, we obtain comparable lower bounds, which likewise rule out exponential savings.
发表机构
- Agentin AI
- University of California, San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。
