Elimination and recursions in the scattering equations

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.

14 pages; v2 references added, minor changes, published version

References in corpus (12)

• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Subleading soft theorem in arbitrary dimension from scattering equations
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• Next to subleading soft-graviton theorem in arbitrary dimensions
• Soft Graviton Theorem in Arbitrary Dimensions
• Scattering Equations and String Theory Amplitudes
• CHY representations for gauge theory and gravity amplitudes with up to three massive particles
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
• Scattering equations and virtuous kinematic numerators and dual-trace functions
• Scattering equations, generating functions and all massless five point tree amplitudes
• Permutation Symmetry of the Scattering Equations
• The Scattering Variety

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• General Solution of the Scattering Equations
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• CHY-Graphs on a Torus
• The Polynomial Form of the Scattering Equations is an H-Basis
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
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• Asymptotic Four Point Functions
• Feynman Rules of Higher-order Poles in CHY Construction
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• Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• A Combinatoric Shortcut to Evaluate CHY-forms
• CHY formula and MHV amplitudes
• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• A differential operator for integrating one-loop scattering equations
• Scattering Equations and a new Factorization for Amplitudes I: Gauge Theories
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• Bootstrapping solutions of scattering equations
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• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• CHY Theory for Several Fields
• Sudakov Representation of the Cachazo-He-Yuan Scattering Equations Formalism