  In this paper, we seek analytically checkable necessary and sufficient
condition for copositivity of a three-dimensional symmetric tensor. We first
show that for a general third order three-dimensional symmetric tensor, this
means to solve a quartic equation and some quadratic equations. All of them can
be solved analytically. Thus, we present an analytical way to check
copositivity of a third order three dimensional symmetric tensor. Then, we
consider a model of vacuum stability for $\mathbb{Z}_3$ scalar dark matter.
This is a special fourth order three-dimensional symmetric tensor. We show that
an analytically expressed necessary and sufficient condition for this model
bounded from below can be given, by using a result given by Ulrich and Watson
in 1994.
