Proudfoot-Speyer degenerations of scattering equations
Overview
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.
The picture below on the left represents the arrangement of four lines in \(\mathbb{R}^{2}\) of Example 2.3. The corresponding reciprocal linear space, on the right, is a cubic surface in \(\mathbb{P}^{3}\). The scattering equations have three solutions represented by the three intersection points of the line with the cubic surface.
Julia source code
ProudfootSpeyerHomotopy.zipMacaulay2 source code
ScatteringEquationsM2.zip
The folder ScatteringEquationsM2.zip contains the following Macaulay2 files:
\(\texttt{ScatteringEquationsUtils.m2}\): auxiliary functions
\(\texttt{TestSolvingScatEqs.m2}\): it solves scattering equations on \(M_{0,5}\) over \(\mathbb{Q}\)
\(\texttt{MacaulayMatrixU34.m2}\): computes the Macaulay matrix in Example 6.3
\(\texttt{CHYamplitude.m2}\): evaluates the CHY amplitude by choosing \(h_1\) and \(h_2\) to be the numerator and denominator of the toric Hessian determinant as described in Example 6.3
The Julia source code is explained in the following Jupyter Notebook tutorial:
The Notebook can be downloaded here tutorial.ipynb