Hankel tensors, Vandermonde tensors and their positivities

An mth order n  -dimensional Hankel tensor is defined as a tensor AA satisfying Ai1…im≡Ai1+⋯+im−mAi1…im≡Ai1+⋯+im−m for some numbers A0,A1,…,Am(n−1)A0,A1,…,Am(n−1). A Hankel tensor possesses a Vandermonde decomposition (VD) A=∑k=1rλkukm where uk=(1,wk,wk2,…,wkn−1)T is called a Vandermonde vector (V-vector). AA is called a Vandermonde tensor (V-tensor) if AA has a VD with each λk=1λk=1. V-tensors are the natural extension of Vandermonde matrices. It is easy to see that all even order V-tensors are positive semidefinite (psd) and thus copositive. An odd order real symmetric tensor is psd only if it is zero. The problem when an odd order Hankel tensor is copositive is open. We present a necessary and sufficient condition for a rank-2 odd-order symmetric real tensor to be copositive. Some necessary conditions for a general V-tensor to be copositive are also presented. The singularity of V-tensors is also investigated, and we show that a V-tensor AA is singular if its V-rank is less than its dimension. This condition becomes necessary if AA is of odd order.