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The Polynomial Form of the Scattering Equations is an H-Basis

Jorrit Bosma, Mads Sogaard, Yang Zhang

TL;DR

The paper proves that the polynomial form of the CHY scattering equations, encoded by $h_m$ for $1\le m\le n-3$, constitutes a Macaulay $H$-basis for the zero-dimensional ideal they generate. This enables a purely linear-algebraic integrand reduction with a sharp degree bound $d^*=(n-3)(n-4)/2$, and a streamlined computation of global residues via either the Bezoutian duality or a direct $H$-basis approach. A key consequence is that the global residue is determined by a single leading coefficient, consistent with Euler–Jacobi vanishing and the existence of a constant in the dual basis. These results significantly simplify CHY amplitude calculations and hint at deeper algebraic structure, with potential analytic expressions for residues and loop-level generalizations forthcoming.

Abstract

We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gröbner basis, but circumvents the heavy computation of the latter. As an example, we apply the H-basis to prove the conjecture that the dual basis of the polynomial scattering equations must contain one constant term.

The Polynomial Form of the Scattering Equations is an H-Basis

TL;DR

The paper proves that the polynomial form of the CHY scattering equations, encoded by for , constitutes a Macaulay -basis for the zero-dimensional ideal they generate. This enables a purely linear-algebraic integrand reduction with a sharp degree bound , and a streamlined computation of global residues via either the Bezoutian duality or a direct -basis approach. A key consequence is that the global residue is determined by a single leading coefficient, consistent with Euler–Jacobi vanishing and the existence of a constant in the dual basis. These results significantly simplify CHY amplitude calculations and hint at deeper algebraic structure, with potential analytic expressions for residues and loop-level generalizations forthcoming.

Abstract

We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gröbner basis, but circumvents the heavy computation of the latter. As an example, we apply the H-basis to prove the conjecture that the dual basis of the polynomial scattering equations must contain one constant term.

Paper Structure

This paper contains 7 sections, 8 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. H and G(röbner) Bases
  3. The Polynomial Form of the Scattering Equations is an H-basis
  4. Integrand Reduction
  5. Global Residues and the Bezoutian
  6. Global Residues and $H$-basis
  7. Conclusion

Key Result

Theorem 2

$\mathcal{H}\equiv\{f_1,\dots,f_k\}$ is an $H$-basis for the ideal $I=\langle f_1,\dots,f_k\rangle$, provided that $(0,\dots,0)$ is the only simultaneous zero of the initial forms $\operatorname{in}(f_1),\dots,\operatorname{in}(f_k)$.

Theorems & Definitions (14)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5: Global Duality
  • Theorem 6: Euler-Jacobi
  • Theorem 7
  • proof
  • ...and 4 more