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.



Waring rank vs tensor rank of symmetric tensors?
A counterexample in $\mathbb{C}$ is given in [A counterexample to Comon's conjecture](https://arxiv.org/abs/1705.08740):



> 
> 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.](https://www.facebook.com/mpimis.nlalg/posts/458148901201044)


