Data Base of Matroids on Eight and Nine Elements with the Half-plane Property and SOS-Rayleigh

Using the classification given in authors' recent work,  we provide the list of all simple matroids on 8 elements of rank 3 and 4 that

• have the half-plane property (HPP),
• don't have the half-plane property
• it is unknown whether they are SOS-Rayleigh (they have HPP),

and the list of some matroids on 8 elements that are not SOS-Rayleigh (in particular, they are not weakly determinantal).

Moreover, we provide the list of all simple and connected matroids on 9 elements of rank 3 that

•  have the half-plane property,
•  don't have the half-plane property,
•  it is unknown whether they are SOS-Rayleigh.

 We give the list of some simple and connected matroids on 9 elements of rank 4 that

• have the half-plane property,
• don't have the half-plane property,
• it is undetected whether they have the half-plane property,
• are candidates for having the half-plane property (numerical data suggest so).

The matroids are listed in the attached files, in an encoded format first given by Matsumoto et. al. in 2011.

For more information on the construction of the fingerprints of matroids, and references please see "Description.pdf".

Among the files provided, one can find Macaulay2  source code for encoding and decoding matroids, and for sorting them with respect to the properties.