The polynomial form of the scattering equations

The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reformulated in polynomial form. The scattering equations for N particles are shown to be equivalent to a Möbius invariant system of N − 3 equations, h̃ m = 0, 2 ≤ m ≤ N − 2, in N variables, where h̃ m is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Möbius invariance appropriately, yields polynomial equations h m = 0, 1 ≤ m ≤ N − 3, in N − 3 variables, where h m has degree m. The linearity of the equations in the individual variables facilitates computation, e.g. the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the h̃ m and h m . The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.