Primary Decompositions in Lorentz-Covariant Rings

Abstract

An effective strategy for constructing compact representations of integral coefficients in scattering amplitudes is to make their singularity structure manifest. This is naturally achieved by treating them as elements of fraction fields of polynomial quotient rings, where poles correspond to algebraic vanishing loci. Simplification via partial fraction decomposition relies on exploiting the structure of the numerators and is governed, at least in a first approximation, by their behaviour on irreducible codimension-two varieties. These are identifiable through primary decompositions of the associated ideals. Despite the availability of general algorithms, primary decompositions remain computationally challenging and often require tailored strategies. In this work, we extend an approach based on ideal saturation, identify an issue related to the breaking of Lorentz covariance in the choice of ideal generators, and present a solution based on fitting ansätze in special kinematic limits. We also provide substantial improvements to an algorithm for the generation of numerical points in proximity of given varieties and to one for testing primality of unmixed ideals, both of which rely on a semi-numerical procedure for computing independent sets and variety dimensions. Finally, we present new codimension-two primary decompositions for five-point massless amplitudes at higher powers in the dimensional regulator, and for finite remainders with five-point one-mass kinematics, where a new spurious singular locus appears.