The Polynomial Form of the Scattering Equations is an H-Basis

TL;DR

This paper proves that the polynomial form of the scattering equations is an H-basis, enabling more efficient integrand reduction and residue calculations compared to Gr"obner bases.

Contribution

It introduces the H-basis structure for scattering equations, simplifying computational methods in amplitude calculations.

Findings

The polynomial scattering equations form a Macaulay H-basis.

The H-basis simplifies integrand reduction and residue computations.

Confirmed the conjecture about the dual basis containing a constant term.

Abstract

We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gr\"obner basis, but circumvents the heavy computation of the latter. As an example, we apply the H-basis to prove the conjecture that the dual basis of the polynomial scattering equations must contain one constant term.