An Algebraic Approach to the Scattering Equations

TL;DR

This paper introduces an algebraic method using companion matrices from computational algebraic geometry to efficiently compute scattering amplitudes in theoretical physics.

Contribution

It applies the companion matrix method to scattering equations, enabling algebraic and algorithmic computation of amplitudes and revealing new identities and rationality properties.

Findings

Simplifies computation of scattering amplitudes using linear algebra.

Derives new identities in scattering amplitudes.

Shows rationality of the integrand in this formalism.

Abstract

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.