The full physical information of a color-ordered helicity amplitude is in the finite remainder:

$$ \textstyle \mathcal{R}^{\ell-\text{loop}}_{n-\text{point}}(|i⟩_{\alpha}, [i|_{\dot\alpha}) = \sum_i {\color{red}{r_{i}}}(|i⟩_{\alpha}, [i|_{\dot\alpha}) \, {\color{blue}{G_i}}(|i⟩_{\alpha} {\scriptsize\times} [i|_{\dot\alpha}) \, . $$

Challenges:

$\circ$ Compute the $\color{blue}{G_i}\,$, a basis of transcendental functions, e.g. the Pentagon Functions;

$\circ$ Reconstruct the $\color{red}{r_{i}}\,$, functions in the field of fractions over a polynomial quotient ring,

$$ \displaystyle \kern10mm R_{n} = \mathbb{F}\Big[|1⟩_{\alpha}, [1|_{\dot\alpha}, \dots, |n⟩_{\alpha}, [n|_{\dot\alpha} \Big] \Big/ J \qquad \text{with} \qquad J = \text{Ideal} \Big( {\textstyle \sum_{i=1}^n} |i\rangle[ i | \Big) \, , $$

$\phantom{\circ}$ in connection with algebraic geometry, and extended to massive particles (spin-spinors).

Amplitudes we computed: $\kern3mm$ GDL + [Abreu, Campbell, Ellis, Ita, Klinkert, Kuschke, Maitre, Page, Ruf, Seth, Sotnikov, ...]