Copositivity for 3rd-Order Symmetric Tensors and Applications

The strict copositivity of 4th-order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) copositivity of 4th-order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided several analytically sufficient conditions for the copositivity of 3rd-order 2-dimensional (3-dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th-order tensors, the analytically sufficient conditions of copositivity are proved for 4th-order 2-dimensional and 3-dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $${\mathbb {Z}}_3$$ scalar dark matter.