Elimination and recursions in the scattering equations

TL;DR

This paper develops a method using elimination theory to explicitly construct polynomial forms of the scattering equations, providing recursive formulas and determinant representations that facilitate their analysis.

Contribution

It introduces a novel explicit construction of the scattering equations polynomial using elimination theory, including recursive formulas and determinant representations.

Findings

Constructed the (n-3)! order polynomial explicitly

Derived recursive formulas for Sylvester determinants

Expressed solutions in terms of Plücker coordinates

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 B\'ezout type of dimension (n-4)!. We present a recursive formula for the Sylvester determinant. Expansion of the determinants yields expressions in terms of Pl\"ucker coordinates. Elimination of the rest of the variables of the scattering equations is also presented.