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.
answered Oct 22, 2017 at 17:18
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$\begingroup$ Thank you @Carlo! Do you know if they coincide for monomials (special symmetric tensors)? I will look at the paper and some references in it though. $\endgroup$– SMDCommented Oct 22, 2017 at 18:27
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$\begingroup$ @SMD For Waring ranks of monomials, see Carlini, Catalisano, Geramita, The solution to the Waring problem for monomials and the sum of coprime monomials (J. Algebra, 2012) (also arxiv.org/abs/1112.3474). I don't know a reference for the tensor rank of a monomial; I'm not sure if it's known. $\endgroup$ Commented Oct 23, 2017 at 5:41
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$\begingroup$ @SMD Some positive results are: here, Corollary 10; here; here. Each of these shows that the conclusion of Comon's conjecture (equality of tensor rank and symmetric tensor rank, aka Waring rank) holds in some special cases. I don't think that monomials are included in any of these special cases. $\endgroup$ Commented Oct 30, 2017 at 5:46
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5$\begingroup$ Shitov's paper has been published today in SIAM J. Appl. Algebra Geometry: epubs.siam.org/doi/abs/10.1137/17M1131970 $\endgroup$– TadashiCommented Sep 11, 2018 at 23:39