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.