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We develop a polynomial reduction procedure that transforms any gauge fixedCHY amplitude integrand for $n$ scattering particles into a $\sigma$-modulimultivariate polynomial of what we call the $\textit{standard form}$. We showthat a standard form polynomial must have a specific $\textit{ladder type}$monomial structure, which has finite size at any $n$, with highest multivariatedegree given by $(n-3)(n-4)/2$. This set of monomials spans a complete basisfor polynomials with rational coefficients in kinematic data on the support ofscattering equations. Subsequently, at tree and one-loop level, we employ theglobal residue theorem to derive a prescription that evaluates any CHYamplitude by means of collecting simple residues at infinity only. Theprescription is then applied explicitly to some tree and one-loop amplitudeexamples.

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