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arXiv · 1504.05597

A note on the gap between rank and border rank

Jeroen Zuiddam
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Abstract

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al.

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Jeroen Zuiddam. 2015-04-21. A note on the gap between rank and border rank. https://doi.org/10.1016/j.laa.2017.03.015

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