AI 中文总结
研究具有退化几何秩的张量，证明其不能提升的充分准则，刻画由特定轨迹导致退化的张量，并给出源于非线性轨迹的此类张量新例子。
AI 中文摘要
具有退化几何秩的张量对应于一个矩阵线性空间，其中秩至多为\(r\)的某些矩阵轨迹维度意外地大。我们证明了此类张量何时不能提升为更大张量的充分准则，并刻画了由秩\(r = 1\)轨迹导致退化的那些张量。我们还给出了源于非线性轨迹的具有退化几何秩的张量新例子。
英文摘要
A tensor of degenerate geometric rank corresponds to a linear space of matrices in which some locus of matrices of rank at most $r$ has unexpectedly large dimension. We prove a sufficient criterion for when such a tensor cannot be lifted to a larger tensor and characterize those tensors for which the degeneracy is caused by the rank $r=1$ locus. We also provide new examples of tensors with degenerate geometric rank stemming from nonlinear loci.
Comments10 pages
