On the rank of 3×3×3-tensors

Abstract

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U ⊗ V ⊗ W is the minimum dimension of a subspace of U ⊗ V ⊗ W containing τ and spanned by fundamental tensors, i.e. tensors of the form u ⊗ v ⊗ w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U ⊗ V ⊗ W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U ⊗ V ⊗ W when the dimensions of U, V and W are higher.