  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 Moebius invariant system of
$N-3$ equations, $\tilde h_m=0$, $2 \leq m \leq N-2$, in $N$ variables, where
$\tilde h_m$ is a homogeneous polynomial of degree m, with the exceptional
property of being linear in each variable taken separately. Fixing the Moebius
invariance appropriately, yields polynomial equations $h_m=0$, $1 \leq m \leq
N-3$, 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
$\tilde h_m$ 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.
