To find whether a set of reduced density matrixes come from a common multi-party state is a hard and important problem. In this paper, (1) we introduce a method to find out some polytopes in one-party eigenvalue-space which are sufficient conditions of this problem. (2) We point out that there are some relations between the compatible conditions and the entanglement of pure states. And we show this idea more clearly in the three-qubit case. (3) We investigate the relations between the compatibility problem and the invariants of a matrix-set under some groups. Furthermore, we show that it is one of the reasons why the compatibility problems which involve the multi-party density matrixes are much more difficult than the one-party case.