  In this paper, an intuitive mathematical formulation is provided to
generalize the residual entanglement for tripartite systems of qubits (Phys.
Rev. A \textbf{61}, 052306 (2000)) to the tripartite systems in higher
dimension. The spirit lies in the tensor treatment of tripartite pure states
(Phys. Rev. A \textbf{72}, 022333 (2005)). A distinct characteristic of the
present generalization is that the formulation for higher dimensional systems
is invariant under permutation of the subsystems, hence is employed as a
criterion to test the existence of genuine tripartite entanglement.
Furthermore, the formulation for pure states can be conveniently extended to
the case of mixed states by utilizing the kronecker product approximate
technique. As applications, we give the analytic approximation of the criterion
for weakly mixed tripartite quantum states and consider the existence of
genuine tripartite entanglement of some weakly mixed states.
