On non-commutative rank and tensor rank
On non-commutative rank and tensor rank
复制标题
关于非交换秩和张量秩
DOI：
10.1080/03081087.2017.1337058
复制
发表时间：
2016
影响因子：
1.1
通讯作者：
V. Makam
中科院分区：
文献类型：
--
作者：
H. Derksen;V. Makam
摘要研究了线性矩阵的交换秩和非交换秩间的关系。我们给出的例子表明，两个秩比任意接近2。这样的例子可以用来给出给定张量的边界秩下界。Landsberg利用这种技巧给出了当m为偶数时，边界秩张量至多的非平凡方程。他还给出了当m为奇数时，边界秩张量至多的方程。利用张量爆破的凹性，我们证明了特征为0的任意场的奇数m的边界秩张量的非平凡方程。我们还给出了伊万尼奥斯、乔和Subrahmanyam关于正则性引理的另一种证明。
Abstract We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give non-trivial equations for the tensors of border rank at most in if m is even. He also gave such equations for tensors of border rank at most in if m is odd. Using concavity of tensor blow-ups we show non-trivial equations for tensors of border rank in for odd m for any field of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam.