New Method to Extend Macaulay Resultant

If Macaulay matrix is degenerated, then we can not derive the relation between the common zeros and coefficients in polynomial system. In order to overcome this, we present a new method to compute the extended Macaulay resultant. After getting the Macaulay matrix, we use elementary row transformation to reduce the matrix, if there is only one nonzero element in some row and this row is generated by all polynomials in system, then it can be regarded as Macaulay resultant. Furthermore, in order to remove the extraneous in resultant we introduce the companion vector for each row in Macaulay matrix to record the coefficients of polynomials in system. The vector is the coefficients of polynomial that is generated by f₁,...,f_n. We can remove most of extraneous factors in Macaulay resultant. This method can be used more widely than regular Macaulay resultant.