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Suppose we work in an algebraically closed field. Then, do the Waring rank (symmetric tensor rank) and tensor rank of a symmetric tensor coincide in general? Recall that tensor rank is rank with respect to the Segre variety and Waring rank is rank with respect to the Veronese variety.

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A counterexample in $\mathbb{C}$ is given in A counterexample to Comon's conjecture:

We present an example of a symmetric tensor of size 800×800×800 which can be written a sum of 903 simple tensors with complex entries but not as a sum of 903 symmetric simple tensors.

It's a very recent result by Yaroslav Shitov, not yet published but it has survived some scrutiny by the tensor community.

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