Maximum likelihood degrees of small matroids

using LikelihoodDegenerations
using HomotopyContinuation
using CapAndHomalg

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CapAndHomalg v1.1.7
Imported OSCAR's components GAP and Singular_jll
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LoadPackage("images")
SetInfoLevel(InfoWarning,0)

On the main page you can download a directory in which to run this code. It contains /MatroidDB/SimpleMatroidkn.txt for various kn.

The following constructs the realization space of the \(27\)-th matroid of rank \(4\) on \(8\) elements in the database file /MatroidDB/SimpleMatroid48.txt.

Z = RealizedMatroidFromDatabase(4,8,27)

0********0**********0****************************0**************0**0**

Realized matroid in dimensions (4, 8)
4×8 Matrix{Expression}:
 1  0  0  1  0   1   1                    1
 0  1  0  1  0  a1   0                   a4
 0  0  1  1  0   0  a2      a2 + a4 - a4*a2
 0  0  0  0  1   1  a3  1 - a2 - a4 + a4*a2
With equations
Expression[a1 - a4 - a1*a2 - a3*a1 + a3*a4 - a4*a1 + a4*a1*a2]
and 6 vanishing Plucker coordinate(s)

We can describe this space as a parametrization + some equations which must be satisfied (shown above)

Z.dimension

3

Z.degree

3

Z.nonbases

6-element Vector{Vector{Int64}}:
 [1, 2, 3, 4]
 [1, 2, 5, 6]
 [1, 3, 5, 7]
 [2, 4, 6, 8]
 [3, 6, 7, 8]
 [4, 5, 7, 8]

Computing the ML degree of this space involves - producing a witness point on each irreducible component of Z - solving the maximum likelihood equations on that component of Z - output a tuple of maximum likelihood degrees for each component

ml_degree(Z)

Tracking 3 paths... 100%|███████████████████████████████| Time: 0:00:18
  # paths tracked:                  3
  # non-singular solutions (real):  3 (0)
  # singular endpoints (real):      0 (0)
  # total solutions (real):         3 (0)
3 solutions left to classify
Found 3 solutions

1-element Vector{Int64}:
 24

There are 4 simple matroids of rank 3 on 5 elements. Below we compute their maximum likelihood degrees

ML_Degrees = [ml_degree(RealizedMatroidFromDatabase(3,5,i)) for i in 1:4];

**********
Completely parametrized. Returning a single point.
0*********
Completely parametrized. Returning a single point.
0******0**
0000******
Completely parametrized. Returning a single point.

display(ML_Degrees)

4-element Vector{Vector{Int64}}:
 [2]
 [1]
 [1]
 [1]