Rank and Border Rank Additivity of Complex Tensors with a Hankel Summand

Abstract

We prove that the CP rank of a complex Hankel tensor of order $m\ge3$ is additive with that of every complex tensor of the same order. Together with the known equality of CP and symmetric ranks for complex Hankel tensors, this yields symmetric rank additivity whenever the other summand has equal CP and symmetric ranks. We also prove that ordinary and symmetric border ranks coincide and are additive on direct sums of complex Hankel tensors of any common order at least three. Their common value is the sum of the middle binary catalecticant ranks. For order three, we further prove border-rank additivity when the other summand has border rank at most three, or is symmetric with at most five essential variables.