General Solution of the Scattering Equations
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. \Delta_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.
Cite
@article{arxiv.1511.09441,
title = {General Solution of the Scattering Equations},
author = {Louise Dolan and Peter Goddard},
journal= {arXiv preprint arXiv:1511.09441},
year = {2016}
}
Comments
v2 completes the proof that the construction yields \Delta_N for all N, identifies it as the hyperdeterminant of a multidimensional matrix, and proves that the polynomial scattering equations constitute a regular sequence, enabling the Hilbert series of the associated variety to be calculated, 26 pages