  We examine the polynomial form of the scattering equations by means of
computational algebraic geometry. The scattering equations are the backbone of
the Cachazo-He-Yuan (CHY) representation of the S-matrix. We explain how the
Bezoutian matrix facilitates the calculation of amplitudes in the CHY
formalism, without explicitly solving the scattering equations or summing over
the individual residues. Since for $n$-particle scattering, the size of the
Bezoutian matrix grows only as $(n-3)\times(n-3)$, our algorithm is very
efficient for analytic and numeric amplitude computations.
