Graphical Scattering Equations
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1.
Otto-von-Guericke University Magdeburg
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2.
Max Planck Institute for Mathematics in the Sciences
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3.
University of Wisconsin–Madison
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4.
University of Michigan
Contributors
Others:
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1.
Max Planck Institute for Mathematics in the Sciences
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2.
Johannes Kepler University of Linz
Description
This page contains auxiliary material for the paper Graphical Scattering Equations by Barbara Betti, Viktoriia Borovik, Bella Finkel, Bernd Sturmfels, and Bailee Zacovic.
The CHY scattering equations on the moduli space $\mathcal{M}_{0,n}$ play a prominent role at the interface of particle physics and algebraic statistics. We study the scattering correspondence when the Mandelstam invariants are restricted to a fixed graph on $n$ vertices.
Technical info
The lists of 129 and 2328 copious graphs for $n=7$ and $8$ respectively, as well as the data with ML degrees, multidegrees and degrees of logarithmic discriminants (for graphs with a universal vertex) up to $n=8$ vertices are available in the following files:
Data456.pdfData7.pdfData8.pdf
The multidegrees for all 129 and 2328 copious graphs for $n=7$ and $8$ are contained in the following files:
Multidegrees7.pdfMultidegrees8.pdf
The corresponding csv versions of all these tables are available in the attachments.
Most of our functions are written in $\texttt{Julia}$ and collected in the package ScatteringGraphs.zip.
A Jupyter Notebook with examples of its usage is provided in ScatteringGraphsTutorial.ipynb. This includes numerical solving of scattering equations, ML degree and multidegree computation with monodromy, and a verification if a graph is copious.
Notes
A computer-aided proof of the Corollary 8.6 by Manuel Kauers is available at
Other
A detailed explanation how to convert a graph with a universal vertex to appropriate gauge fixing to compute the degree of the logarithmic discriminant is available in the Jupyter Notebook in the folder Discriminants.zip. There, it is done for all copious graphs with a universal vertex for $n=8$.
A Macaulay2 code verifying Proposition 4.11 is available in the M2.zip folder.
Files
Data6.csv
Files
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Additional details
Related works
- Is supplement to
- Preprint: arXiv:2511.07316v2 (arXiv)
Funding
Dates
- Created
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2025-11-11