Computing Algebraic Matroids

Symbolic Algebra:

We have implemented the symbolic algorithm in Macaulay2. The code is in the file matroids.m2.

Linear Algebra and Numerical Algebraic Geometry:

This algorithm breaks up into a few steps:

1. Generate a numerical Jacobian matrix, in one of the following ways:
   1. Compute a Jacobian symbolically, and then specialize to a generic point of the variety. A generic point can be obtained for a parametrization by choosing random values for the parameters, and for a variety defined by its ideal using Bertini Tracktype:1.
   2. Compute the Jacobian numerically using Bertini: Run Tracktype:1, save a witness point from main_data in start, and run Tracktype:-3.
2. Use numerical linear algebra, in Sage, Maple, or MATLAB, to compute a list of bases and circuits from the Jacobian matrix.
3. Run Bertini, Tracktype:5 on a list of projection vectors and extract the degree of the projection for the circuit degree, and the degree of the fiber for the base degree. Sample Bash scripts for this are: decBases.sh and decCircs.sh.