Hankel Tensor Decompositions and Ranks

Hankel tensors are generalizations of Hankel matrices. This article studies both the computational and algebraic aspects of Hankel tensor ranks. We prove that for a low rank symmetric tensor, there exists a base change to make it a Hankel tensor. We also provide an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional Hankel tensor of even order or order three, we prove that the candecomp-parafac rank, symmetric rank, Vandermonde rank, border rank, symmetric border rank, and Vandermonde border rank all coincide with each other. Some open questions are also posed.