What can algebraic geometry do for rational functions in scattering amplitudes?

Scattering amplitudes are weighted sums of transcendental functions with rational-function coefficients. The latter arise, for example, from tensor and integration-by-parts reductions, and are naturally expressed in spinor-helicity variables: elements of the field of fractions of a polynomial quotient ring. In least-common-denominator form they can have numerators containing millions of monomials, while obtaining even a single exact numerical sample can be computationally expensive.

I will discuss how computational algebraic geometry can help expose and exploit their underlying structure, making reconstruction of analytic expressions from numerical samples feasible. In particular, I will describe multivariate partial-fraction decompositions constructed from interpolation on a two-dimensional slice, as used in a recent two-loop Higgs+2jet calculation, or through p-adic probes near codimension-two irreducible varieties. The latter approach is closely connected to primary decomposition, and I will discuss practical approaches for their computation in Lorentz-covariant kinematic rings. If time permits, I will discuss upcoming work on factorization properties of spinor-helicity rings and corresponding factorization algorithms.