On the ranks of the third secant variety of Segre–Veronese embeddings

Abstract

We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors $T\in V_1\otimes \cdots \otimes V_k$ (Segre case) we can show that all ranks among 3 and $k-1$ arise and if $dim{V}_i\ge 3$ for all i’s, then all the ranks between k and $2k-1$ arise.

Keywords:

• Partially symmetric tensor
• border rank
• tensor rank

AMS Subject Classification:

• 15A69

Acknowledgements

We want to thank the anonymous referee and the Handling Editor Jan Draisma for their careful jobs that improved the presentation of this paper. A special thank to the referee for her/his very interesting questions that encouraged us in giving a better version of Theorem 1.8.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Gnsaga of INDAM.