Scattering amplitudes over finite fields and multivariate functional reconstruction

TL;DR

This paper introduces an algorithm for reconstructing multivariate polynomials and rational functions from finite field evaluations, enabling efficient calculations in computer algebra and scattering amplitude computations.

Contribution

It presents a novel reconstruction algorithm applicable to multivariate polynomials and rational functions, optimized for use in scattering amplitude calculations.

Findings

Efficient reconstruction of multivariate functions from finite field evaluations.

Successful application to complex scattering amplitude calculations in Yang-Mills theory.

Scales well with the number of variables and problem complexity.

Abstract

Several problems in computer algebra can be efficiently solved by reducing them to calculations over finite fields. In this paper, we describe an algorithm for the reconstruction of multivariate polynomials and rational functions from their evaluation over finite fields. Calculations over finite fields can in turn be efficiently performed using machine-size integers in statically-typed languages. We then discuss the application of the algorithm to several techniques related to the computation of scattering amplitudes, such as the four- and six-dimensional spinor-helicity formalism, tree-level recursion relations, and multi-loop integrand reduction via generalized unitarity. The method has good efficiency and scales well with the number of variables and the complexity of the problem. As an example combining these techniques, we present the calculation of full analytic expressions for the two-loop five-point on-shell integrands of the maximal cuts of the planar penta-box and the non-planar double-pentagon topologies in Yang-Mills theory, for a complete set of independent helicity configurations.