Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes

We develop a polynomial reduction procedure that transforms any gauge fixed chy amplitude integrand for n scattering particles into a σ-moduli multivariate polynomial of what we call the standard form. we show that a standard form polynomial must have a specific ladder type monomial structure, which has finite size at any n, with highest multivariate degree given by (n − 3)(n − 4)/2. this set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. subsequently, at tree and one-loop level, we employ the global residue theorem to derive a prescription that evaluates any chy amplitude by means of collecting simple residues at infinity only. the prescription is then applied explicitly to some tree and one-loop amplitude examples.