An Upper Bound for the Tensor Rank

Let be a tensor of format , , over . We prove that has tensor rank at most . 1. Introduction Fix integers and , , and an algebraically closed base field . Let be a tensor of format over . The tensor rank of is the minimal integer such that with (see [1–6]). Classical papers (e.g., [7]) continue to suggest new results (see [8]). Let be the maximum of all integers , . In this paper we prove the following result. Theorem 1. For all integers and one has . This result is not optimal. It is not sharp when , since by elementary linear algebra. For large the bound should be even worse. In our opinion to get stronger results one should split the set of all into subregions. For instance, we think that for large the cases with and the cases with are quite different. We make the definitions in the general setting of the Segre-Veronese embeddings of projective spaces (i.e., of partially symmetric tensors), but we only use the case of the usual Segre embedding, that is, the usual tensor rank. The tensor has zero as its tensor rank. If , then the tensors and have the same rank. Hence it is sufficient to study the function “tensor rank” on the projectivisation of the vector space . We may translate the tensor rank and the integer in the following language. For each subset of a projective space, let denote the linear span of . For each integral variety and any the -rank of is the minimal cardinality of a finite set such that . Now assume . The maximal -rank is the maximum of all integers , . Fix integers , , , and , . Set . Let , be the Segre-Veronese embedding of multidegree , that is, the embedding of induced by the -vector space of all polynomials , , , whose nonzero monomials have degree with respect to the variables , . Set . The variety is the Segre embedding of . Fix , . Let be the set of all nonzero tensors of format associated with . We have . Hence . To prove Theorem 1 we refine the notion of -rank in the following way. Definition 2. Fix positive integers , , , and , . A small box of is a closed set with being a hyperplane of for all . A large box of is a product such that there is with being a hyperplane, while for all . A small polybox (resp., large polybox) of is a finite union of small (resp., large) boxes of . A small box (resp., small polybox, resp., large box, resp., large polybox) is the image by of a small box (resp., small polybox, resp., large box, resp., large polybox) of . Definition 3. Fix positive integers , , , and , , and set . Fix . The rank of is the minimal cardinality of a finite set such that . The unboxed rank (resp., small unboxed rank)