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Proudfoot-Speyer degenerations of scattering equations

Barbara Betti, Viktoriia Borovik, Simon Telen

TL;DR

This work develops an algebraic-geometry framework for solving scattering equations arising from hyperplane arrangements by viewing them as linear equations on a reciprocal linear space ${\cal R}_L$ and solving them via a Proudfoot-Speyer Gröbner-degeneration-based homotopy. It characterizes the solution count through topological and matroid invariants, notably $(-1)^d\chi(X)$ and $\deg{\cal R}_L$, and compares reciprocal and ML degrees using a flats-based stratification. In the CHY setting on $\mathcal{M}_{0,m}$, the authors compute $\deg{\cal R}_{L_m}=(m-3)(m-3)!$, describe boundary contributions from type (ii) flats, and propose a multiplicity conjecture that yields a decomposition of the solution set into open and boundary components. The paper provides practical computational tools, including a Julia implementation of the Proudfoot-Speyer homotopy, and demonstrates their application to CHY scattering amplitudes, offering new insights into when algebraic-statistical and geometric perspectives align for these rational systems.

Abstract

We study scattering equations of hyperplane arrangements from the perspective of combinatorial commutative algebra and numerical algebraic geometry. We formulate the problem as linear equations on a reciprocal linear space and develop a degeneration-based homotopy algorithm for solving them. We investigate the Hilbert regularity of the corresponding homogeneous ideal and apply our methods to CHY scattering equations.

Proudfoot-Speyer degenerations of scattering equations

TL;DR

This work develops an algebraic-geometry framework for solving scattering equations arising from hyperplane arrangements by viewing them as linear equations on a reciprocal linear space and solving them via a Proudfoot-Speyer Gröbner-degeneration-based homotopy. It characterizes the solution count through topological and matroid invariants, notably and , and compares reciprocal and ML degrees using a flats-based stratification. In the CHY setting on , the authors compute , describe boundary contributions from type (ii) flats, and propose a multiplicity conjecture that yields a decomposition of the solution set into open and boundary components. The paper provides practical computational tools, including a Julia implementation of the Proudfoot-Speyer homotopy, and demonstrates their application to CHY scattering amplitudes, offering new insights into when algebraic-statistical and geometric perspectives align for these rational systems.

Abstract

We study scattering equations of hyperplane arrangements from the perspective of combinatorial commutative algebra and numerical algebraic geometry. We formulate the problem as linear equations on a reciprocal linear space and develop a degeneration-based homotopy algorithm for solving them. We investigate the Hilbert regularity of the corresponding homogeneous ideal and apply our methods to CHY scattering equations.
Paper Structure (7 sections, 18 theorems, 35 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 7 sections, 18 theorems, 35 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scattering equations, reciprocal linear spaces and maximum likelihood
  3. Reciprocal versus ML degree
  4. Proudfoot-Speyer homotopies
  5. Scattering equations on $\mathcal{M}_{0,m}$
  6. Hilbert regularity
  7. Acknowledgements.

Key Result

Theorem 1.2

For generic $L \in \mathbb{C}^{(d+1) \times (n+1)}$ and generic $u \in \mathbb{C}^{n+1}$, the Proudfoot-Speyer homotopy described in Section sec:4 is optimal, meaning that the number of solution paths tracked equals the number of solutions to eq:scatteringeqs in $X$.

Figures (2)

  • Figure 1: An arrangement of four lines in $\mathbb{R}^2$ whose scattering equations have three solutions and whose reciprocal linear space is a cubic surface in $\mathbb{P}^3$.
  • Figure 2: Hasse diagram of type (ii) flats $I_r(W)$ of $M(L_6)$.

Theorems & Definitions (47)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Example 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Example 2.6
  • Example 2.7
  • ...and 37 more