Certifying Maximum Likelihood Degrees of Matroid Strata

We present the result of the certification method described in [1] for the computation of the ML degree for matroids of rank k on m elements. The main theorem in Section 4 of [1], for fixed k, m, illustrates the matroid strata by dimension, and all occurring ML degrees together with their multiplicity of occurrences.

In each folder 'CertificatesAndSummaries'*k*m, for each simple rank k matroid on m atoms indexed by the database we used (with the exceptions of (3,9) #1#2#3#5 and (4,8) #1#2) there are two files: MatroidSummary_i and Certificate_i.

MatroidSummary_i summarizes the certification attempts we made - that is, how many, and what the certified lower bound for the ML degree was. An instance of a certification which obtained the maximum bound is saved in Certificate_i.
    
For some matroids, we had the resources to run the certification process multiple times, whereas for matroids with large ML degrees, we only ran it once. Nonetheless, each certificate produces a lower bound for the ML degree of the corresponding matroid. 
If an additional index exists, this indicates that the realization space of the matroid had multiple irreducible components (see (4,8) # 160 which has two components) and each file corresponds to the above process for a single irreducible component. 

Finally, the file Certificate_48 certifies the lower bound of the Euler characteristic of the space X(4,8) discussed in Section 6 of [1].

References

[1] D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels, and S. Telen: Likelihood Degenerations, arXiv:2107.10518.

[2] P. Breiding, K. Rose, and S. Timme: Certifying zeros of polynomial systems using interval arithmetic, arXiv:2011.05000.