超越线性展平障碍的显式张量
Explicit tensors beyond the linear flattening barrier
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- Concordia University（康科迪亚大学）
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中文总结 AI 辅助
本文构造了边界秩至少为3n-o(n)的显式张量，并证明线性展平方法无法超越2n-1的下界，从而提供了超越线性展平障碍的显式示例。
中文摘要 AI 辅助
我们构造了边界秩至少为3n-o(n)的显式n x n x n张量，改进了(Landsberg和Michalek 2025)所创的(2 + $\varepsilon$)n的先前纪录。我们还证明了线性展平方法不能用于建立超过2n-1的边界秩下界。当n为奇数时，我们证明该下界可由Koszul展平实现。当n为偶数时，(Landsberg 2015)的一个结果表明Koszul展平可以达到2n-2，留下大小为1的开放间隙。我们的2n-1下界改进了(Garg等人2019)和(Buczyński 2026)所提出的最佳已知6n-4线性展平障碍。综合这些结果，表明我们的3n-o(n)构造以及Landsberg和Michalek的构造，提供了比任何线性展平所能达到的边界秩更高的显式张量示例。
英文摘要
We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + $\varepsilon$)n due to (Landsberg and Michalek 2025). We also prove that the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n-1. When n is odd, we prove that this bound is achieved by Koszul flattenings. When n is even, a result of (Landsberg 2015) shows that Koszul flattenings can achieve 2n-2, leaving an open gap of size one. Our 2n-1 bound improves the best known 6n-4 linear flattening barrier due to (Garg et al. 2019) and (Buczyński 2026). Combined, these results show that our 3n-o(n) construction, as well as the construction of Landsberg and Michalek, provide explicit examples of tensors with higher border rank than any linear flattening can achieve.
