The Entry Formula for a Mixed Resultant Matrix of Three-Variable Polynomial System

Resultant matrix is one of important tools in elimination theory. one of many resultant matrices is mixed Cayley-Sylvester resultant matrix. This paper starts from the definition of mixed Cayley-Sylvester resultant matrices, for a three-variable polynomial system, and use the property of determinant to simplify the main steps to avoid polynomial division and to improve the computational efficiency of mixed Cayley-Sylvester resultant.