The M-theory S-matrix

We obtain compact formulae for tree superamplitudes for the massless sector of M-theory, i.e., 11 dimensional supergravity, as well as 10 dimensional supersymmetric Yang-Mills, gravity and Born-Infeld. These are based on the \emph{polarised scattering equations} which incorporate polarization data into a spinor field on the Riemann sphere and arise from a twistorial representation of ambitwistor strings in 10 and 11 dimensions. They naturally extend amplitude formulae to manifest maximal supersymmetry. The framework is the natural generalization of twistorial ambitwistor string formulae found previously in four and six dimensions.