Scattering Equations and Global Duality of Residues

Abstract: 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.