超越弱边界对偶性的边界秩下界
Border rank lower bounds beyond weak border apolarity
- Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文首次展示超越弱边界对偶性的边界秩下界示例，证明某些张量的理想列表虽非空但无一满足全部条件，涵盖矩阵乘法相关张量及群论相关案例。
AI中文摘要：
我们提供了首批利用边界对偶性（border apolarity）获得的边界秩下界示例，这些示例超越了“弱边界对偶性”（weak border apolarity）。由Buczyńska-Buczyński引入的边界对偶性将边界秩大于$r$的张量刻画为其零化子不包含一个具有某些依赖于$r$和张量所在空间的特定性质的理想。先前边界对偶性的应用依赖于Conner-Harper-Landsberg开发的算法。对于具有大对称群的张量，该算法列出一组满足上述部分性质的理想。我们给出了首批列表非空但能够证明该列表中没有任何理想满足边界对偶性所要求的所有必要条件的示例。新的下界示例包括复杂性研究中受关注的张量：第一个是边界秩此前未知的矩阵乘法张量的Schönhage三重和，以及两个张量副本的直和，该张量可能被用于证明矩阵乘法指数为2。第三个示例与A. Leitner的工作相关，在群论中具有研究价值，我们也对其进行了分析。
英文摘要：
We provide first examples of border rank lower bounds obtained using border apolarity beyond ``weak border apolarity''. Border apolarity introduced by Buczyńska-Buczyński characterizes tensors of border rank greater than $r$ as those whose annihilator does not contain an ideal with certain properties depending on $r$ and the ambient space of the tensor. Previous applications of border apolarity depended on the algorithm developed by Conner-Harper-Landsberg. For a tensor with a large symmetry group it lists a set of ideals that satisfy some of the properties mentioned above. We give first examples where the list is non-empty but we are able to prove that no ideal on this list satisfies all necessary conditions coming from border apolarity. The new lower bound examples include tensors of interest in complexity: the first Schönhage triple sum of matrix multiplication tensors where the border rank was unknown, and the direct sum of two copies of a tensor that could potentially be used to prove the exponent of matrix multiplication is two. A third example of interest in group theory related to work of A. Leitner is also analyzed.
