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# On Strassen's rank additivity for small three-way tensors

journal contribution
posted on 06.08.2019, 08:31 by Jarosław Buczyński, Elisa Postinghel, Filip Rupniewski
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 ${\mathbb C}^k\otimes {\mathbb C}^3\otimes {\mathbb C}^3$ 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 the 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 ${\mathbb C}^4\otimes {\mathbb C}^4\otimes {\mathbb C}^4$. Some of our results are valid over an arbitrary base field.

• Science

## Department

• Mathematical Sciences

## Published in

SIAM Journal on Matrix Analysis and Applications

41

1

106-133

## Publisher

Society for Industrial and Applied Mathematics (SIAM)

## Version

VoR (Version of Record)

## Rights holder

© Society for Industrial and Applied Mathematics

17/07/2019

2020-01-14

2020

0895-4798

1095-7162

en

## Depositor

Dr Elisa Postinghel

## Exports

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