  We develop a polynomial reduction procedure that transforms any gauge fixed
CHY amplitude integrand for $n$ scattering particles into a $\sigma$-moduli
multivariate polynomial of what we call the $\textit{standard form}$. We show
that a standard form polynomial must have a specific $\textit{ladder type}$
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.
