Partially symmetric tensor rank: the description of the non-uniqueness case for low rank

Abstract

We study the partially symmetric tensors T with low rank whose rank is not achieved by a unique set when the rank has an upper bound like , where is the minimum of the entries of the multidegree of T. For the proof, we use the Hilbert function of finite subsets of a product of projective spaces with respect to a Segre–Veronese embedding.

Keywords:

• Partially symmetric tensor
• Segre–Veronese variety
• tensor rank
• X-rank

AMS Subject Classifications:

• 15A69
• 14N05

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The author was partially supported by MIUR  and GNSAGA of INdAM   (Italy).