General containment criterion for realizable matroid varieties

Determine whether, for arbitrary realizable rank-three matroids \(M\) and \(N\) on the same ground set, the containment \(V_N\subset V_M\) holds, and characterize the conditions equivalent to this containment.

Background

The appendix studies how to remove redundant components from decompositions of circuit varieties by deciding containment relations between matroid varieties. The paper proves that N≥MN\geq M in dependency order is necessary for VN⊂VMV_N\subset V_M when MM and NN are realizable rank-three matroids. It then develops perturbation arguments that establish containment for particular families, but it does not resolve the general containment question.

References

Given realizable matroids $M$ and $N$ of rank three on the same ground set, is $V_{N}\subset V_{M}$?

— Minimal matroids in dependency posets: algorithms and applications to computing irreducible decompositions of circuit varieties  (2502.00799 - Liwski et al., 2 Feb 2025) in Appendix, Section \"Techniques for verifying redundancy\", Question~\ref{question}