Unbounded Comon rank gaps for third-order symmetric tensors

Construct over the real or complex numbers a family of third-order symmetric tensors with unbounded Comon rank gaps, either by taking direct sums or by another method.

Background

The paper constructs third-order symmetric tensors over both the real and complex fields whose Comon rank gaps are two and three. Specifically, direct sums of two and three independent copies of Lovitz’s 27-dimensional tensor produce gaps of two and three, respectively.

The conclusion asks whether this phenomenon can continue without bound. The unresolved problem is to determine whether a family of third-order symmetric tensors exists whose symmetric rank exceeds its CP rank by arbitrarily large amounts, either through further direct-sum constructions or through a different construction method.