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General Solution of the Scattering Equations

Louise Dolan, Peter Goddard

TL;DR

The paper provides an explicit algebraic solution to the massless scattering equations by reformulating them as a zero-dimensional, polynomial system with $N-3$ equations $h_m$, each linear in the $z_a$ variables. It shows how elimination yields a single polynomial $\Delta_N(u/v)$ of degree $\delta_N=(N-3)!$, identified as the sparse resultant (hyperdeterminant) of the system, and proves that the solution set has exactly $\,(N-3)!$ points via a regular sequence and Hilbert-series analysis. The authors construct concrete Δ_N polynomials for $N=4$–$7$ and provide a general determinant-based framework for arbitrary $N$, together with a leading-power analysis that guarantees nontriviality under generic kinematics. The work connects the scattering equations to deep algebraic concepts—sparse resultants, hyperdeterminants, and GKZ theory—highlighting the regularity of the defining ideal and offering a robust algebraic handle on CHY tree amplitudes for massless theories.

Abstract

The scattering equations, originally introduced by Fairlie and Roberts in 1972 and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension, have been reformulated in polynomial form. The scattering equations for N particles are equivalent to N-3 polynomial equations h_m=0, m=1,...,N-3, in N-3 variables, where h_m has degree m and is linear in the individual variables. Facilitated by this linearity, elimination theory is used to construct a single variable polynomial equation of degree (N-3)! determining the solutions. Δ_N is the sparse resultant of the system of polynomial scattering equations and it can be identified as the hyperdeterminant of a multidimensional matrix of border format within the terminology of Gel'fand, Kapranov and Zelevinsky. Macaulay's Unmixedness Theorem is used to show that the polynomials of the scattering equations constitute a regular sequence, enabling the Hilbert series of the variety determined by the scattering equations to be calculated, independently showing that they have (N-3)! solutions.

General Solution of the Scattering Equations

TL;DR

The paper provides an explicit algebraic solution to the massless scattering equations by reformulating them as a zero-dimensional, polynomial system with equations , each linear in the variables. It shows how elimination yields a single polynomial of degree , identified as the sparse resultant (hyperdeterminant) of the system, and proves that the solution set has exactly points via a regular sequence and Hilbert-series analysis. The authors construct concrete Δ_N polynomials for – and provide a general determinant-based framework for arbitrary , together with a leading-power analysis that guarantees nontriviality under generic kinematics. The work connects the scattering equations to deep algebraic concepts—sparse resultants, hyperdeterminants, and GKZ theory—highlighting the regularity of the defining ideal and offering a robust algebraic handle on CHY tree amplitudes for massless theories.

Abstract

The scattering equations, originally introduced by Fairlie and Roberts in 1972 and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension, have been reformulated in polynomial form. The scattering equations for N particles are equivalent to N-3 polynomial equations h_m=0, m=1,...,N-3, in N-3 variables, where h_m has degree m and is linear in the individual variables. Facilitated by this linearity, elimination theory is used to construct a single variable polynomial equation of degree (N-3)! determining the solutions. Δ_N is the sparse resultant of the system of polynomial scattering equations and it can be identified as the hyperdeterminant of a multidimensional matrix of border format within the terminology of Gel'fand, Kapranov and Zelevinsky. Macaulay's Unmixedness Theorem is used to show that the polynomials of the scattering equations constitute a regular sequence, enabling the Hilbert series of the variety determined by the scattering equations to be calculated, independently showing that they have (N-3)! solutions.

Paper Structure

This paper contains 17 sections, 93 equations.

Table of Contents

  1. Introduction
  2. Review of the Cases $N=4,5$ and $6$
  3. Solutions for $N=4$ and $N=5$.
  4. Solution for $N=6$.
  5. Direct demonstration that $\Delta_6$ is the required sextic.
  6. Determination of $x,y$ in terms of $u,v$ satisfying $\Delta_6=0$.
  7. Elimination Theory for $N=7$
  8. Construction of the 24th order polynomial $\Delta_7$ using elimination theory
  9. Direct demonstration that $\Delta_7$ is the required polynomial of degree 24.
  10. Elimination Theory for General $N$
  11. Construction of the $(N-3)!$-th degree polynomial $\Delta_N$ using elimination theory
  12. Direct demonstration that $\Delta_N$ is the required polynomial.
  13. The Leading Power of $v$, $v^{\delta_N}$.
  14. Resultants, Hyperdeterminants and Hilbert Series
  15. Sparse Resultants
  16. ...and 2 more sections