Copositivity For 3rd Order Symmetric Tensors And Applications

The main purposes of this paper focus on discussing the analytical expressions of the copositivity for 3rd and 4th order symmetric tensors,and finding the analytical vacuum stability conditions for vacuum stability for Z3 group scalar dark matter.Chapter 1,we give a introduction which mainly introduces the basic concepts and research status of copositive tensors,and then we expound the concept and development of Z3 group scalar dark matter,and further explain the main work of this paper.Chapter 2,we mainly introduce homogenous polynomials of 3rd and 4th order symmetric tensor and achieve the sufficient conditions for copositivity of 3rd order symmetric tensors.By using the method of reducing orders or dimensions of tensor,we also obtain the analytical expressions of(strict)copositivity of 4th order symmetric tensors.Chapter 3,we present that the vacuum stability of Z3 group scalar dark matter V(h1,h2,s)is equivalent to the strict copositivity of the coupling tensor ?=(?ijkl),then apply the analytical expressions of(strict)copositivity of 4th order symmetric tensors to the vacuum stability of scalar dark matter in Z3 discrete group.The analytical expression of vacuum stability is given,which is different from other literatures.