Elimination and recursions in the scattering equations

TL;DR

This work applies classical elimination theory to the polynomial form of the CHY scattering equations to produce a single-variable polynomial of degree $(n-3)!$ in one puncture variable. It provides two determinantal representations of the resultant: a Sylvester-type determinant of size $(n-3)!$ and a Bézout-type determinant of size $(n-4)!$, with a recursive construction that reveals an expansion in Plücker coordinates and a connection to Cayley hyperdeterminants. The authors work through explicit cases $n=6,7,8$ and present a general framework, including a recursion for the determinant and a method to eliminate all remaining variables. They also discuss how these eliminations enable direct evaluation of sums over solutions (amplitudes) without solving the equations, using companion matrices and Vieta relations. The results offer structural insights and potential computational simplifications for CHY amplitudes and suggest connections to triangulations of Grassmannians and to recursion relations in scattering theory.

Abstract

We use the elimination theory to explicitly construct the (n-3)! order polynomial in one of the variables of the scattering equations. The answer can be given either in terms of a determinant of Sylvester type of dimension (n-3)! or a determinant of Bézout type of dimension (n-4)!. We present a recursive formula for the Sylvester determinant. Expansion of the determinants yields expressions in terms of Plücker coordinates. Elimination of the rest of the variables of the scattering equations is also presented.