We address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in $C^k \otimes C^3 \otimes C^3$ for any $k$, then the additivity also holds. More generally, if one of the tensors is concise and its rank is at most 2 more than the dimension of one of the linear spaces, then additivity holds. In addition we also treat some cases of the additivity of border rank of such tensors. In particular, we show that the additivity of the border rank holds if the direct sum tensor is contained in $C^4 \otimes C^4 \otimes C^4$.

中文翻译：

---

关于小三向张量的 Strassen 秩可加性

我们解决了张量等级的可加性问题。这是对于我们研究的两个独立张量，如果它们的直和的秩等于它们各自的秩的总和。在 Shitov 提出最近的反例之前，对这个问题的肯定答案以前被称为 Strassen 猜想。后者不是很明确，并且只知道它们在非常大的张量空间中渐近存在。在本文中，我们证明了对于一些小的三向张量，可加性成立。例如，如果张量之一的秩最多为 6，则可加性成立。或者，如果对于任何 $k$，其中一个张量存在于 $C^k \otimes C^3 \otimes C^3$ 中，则可加性也成立。更一般地，如果张量之一是简洁的，并且它的秩最多比线性空间之一的维数多 2，那么可加性成立。此外，我们还处理了一些此类张量边界秩可加性的情况。特别地，我们表明，如果直和张量包含在 $C^4 \otimes C^4 \otimes C^4$ 中，则边界秩的可加性成立。