Copositivity of a class of fourth order symmetric tensors

摘要: In this paper, we mainly dicuss  the analytic conditions of copositivity of  a class of 4th order 3-dimensional symmetric tensors.  For a 4th order 3-dimensional symmetric tensor with its entries  $1$ or $-1$, an analytic  necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. And by means of usual maxi-min theory,  a  necessary and sufficient condition is established for copositivity of such a  tensor also.  Applying these conclusions to a general 4th order 3-dimensional symmetric tensor,  the analytic conditions  are successfully obtained  for  verifying the (strict) copositivity, and these conditions can be very easily parsed and validated.