Strict Copositivity for a Class of 3rd Order Symmetric Tensors

In this article, we mainly give the strictly copositive conditions of a special class of third order three dimensional symmetric tensors. More specifically, by means of the polynomial decomposition method,  the analytic  sufficient and necessary  conditions are established for checking the strict  copositivity of a 3rd order 3-dimensional symmetric tensor with its entries  in $\{-1,0,1\}$.   Several  strict inequalities of cubic ternary homogeneous polynomials are presented by applying these conclusions. Some criteria which ensure the strict  copositivity of a general 3rd order 3-dimensional tensor are obtained