Question: resultant of multivariate polynomials

I am trying to solve the so called Bethe ansatz equations in Mathematical Physics.

I have four multivariate polynomials equations with four unknown p_i(x[1],x[2],x[3],x[4]) i=1,2,3,4; which have integer coefficients. The equations are mixed; P_i is linear in x[j] for x[j]\ne x[i]; and x[i] is of order 10. Due to this structure,  I first eliminate x[4] from these equations. Then use resultant to eliminate x[3],  I got error messages (invalid series, in gcd), which I could not understand. The I tried LinearAlgebra to calculate SylvesterMatrix, then its determinants. I again got error messages (invalid series). So I tried to used Determinant(A, method=algnum), which seemed very slow. When I tried the BezoutMatrix, the output exceed the maple limit of 10....00.

Any sugguestions? I have read the paper by Carlos D'Andrea, but did find it confusing, with too many unfamiliar terminologies.