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.
Highlights
That there are only N − 3 linearly independent equations in the system
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 hm = 0, 1 ≤ m ≤ N − 3, in N − 3 variables, where hm has degree m and is linear in the individual variables
Summary
4.1 Construction of the (N − 3)!-th degree polynomial ∆N using elimination theory. For N ≥ 5, write xa = za+1, 1 ≤ a ≤ N − 4, u = zN−2, v = zN−1. Dividing CNj −4 and RNj −4 into subsets CNj,k−4 ∼= xkN−k−3CNj−−k5 and RNj,k−4 ∼= xkN−k−3RNj−−k5, respectively, and subdividing further and so on inductively defines orders on CNj −4 and RNj −4, with respect to which φ is lower triangular, and, reordering the rows and columns of M (N) brings it into a form which is lower triangular It follows that det M (N), which contains the term in ∆N involving vδN , is given, up to sign, by a single product of the (N − 3)!
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