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)×(n-3), our algorithm is very efficient for analytic and numeric amplitude computations.
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Søgaard et al. (2016) studied this question.
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