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, 1 ≤ m ≤ 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, Δ N = 0, 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.
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Dolan, L., Goddard, P. General solution of the scattering equations. J. High Energ. Phys. 2016, 149 (2016). https://doi.org/10.1007/JHEP10(2016)149
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DOI: https://doi.org/10.1007/JHEP10(2016)149