论文标题
张张量排名$ 2 \ times 2 $矩阵乘法张量的直接总和为14
Tensor rank of the direct sum of two copies of $2 \times 2$ matrix multiplication tensor is 14
论文作者
论文摘要
本文涉及张量排名的添加性问题。这是针对两个独立的张量，当它们的直接总和等于其各个等级的总和时。该声明说，添加性总是以前被称为Strassen的猜想（1969），直到Shatov提出了反示例（2019）。它们不是显式的，并且仅在非常大的张量空间中渐近地存在。在本文中，我们表明，对于一些添加性的一些小型三向张量。例如，我们给出证明Strassen（1969）所说的另一个猜想是正确的。这是一般Strassen的添加性猜想的特殊情况，其中张量是一对$ 2 \ times 2 $矩阵乘法张量。此外，我们表明Alexeev-Forbes-Tsimerman替代方法保留了直接张紧器的结构。
The article is concerned with the problem of the additivity of the tensor rank. That is for two independent tensors we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we show that for some small three-way tensors the additivity holds. For instance, we give a proof that another conjecture stated by Strassen (1969) is true. It is the particular case of the general Strassen's additivity conjecture where tensors are a pair of $2 \times 2$ matrix multiplication tensors. In addition, we show that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.
