Positive Definiteness of $4$th Order $3$-Dimensional Symmetric Tensors with entries $-1$, $0$, $1$

摘要:  It is well-known that a symmetric matrix with its entries $\pm1$ is not positive definite.  But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly  dicuss  the positive (semi-)definiteness criterion of a class of $4$th order $3$-dimensional symmetric tensors  with entries $t_{ijkl}\in\{-1,0,1\}$. Through theoretical derivations and detailed classification discussions, the criterion for determining the positive (semi-)definiteness of such a class of tensors   are provided based on the relationships and number values of its entries. Which establishes some unique properties of higher  symmetric tensors that distinct from ones of matrces