Copositivity for a class of fourth order symmetric tensors

In this paper, we mainly discuss the analytic expression of exact copositivity of 4th order symmetric tensor defined by the special physical model. We first show that for the general 4th order 2-dimensional symmetric tensor, it can be transformed into solving the quadratic polynomials, and then we give a necessary and sufficient condition to test the copositivity of 4th order 2-dimensional symmetric tensor. Based on this, we consider a special 4th order 3-dimensional symmetric tensor defined by the vacuum stability for $\mathbb{Z}_{3}$ scalar dark matter, and obtain the necessary and sufficient condition for its copositivity.