Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties

Abstract

Matroid theory provides a unifying framework for studying dependence across combinatorics, geometry, and applications ranging from rigidity to statistics. In this work, we study circuit varieties of matroids, defined by their minimal dependencies, which play a central role in modeling determinantal varieties, rigidity problems, and conditional independence relations. We introduce an efficient computational strategy for decomposing the circuit variety of a given matroid $M$, based on an algorithm that identifies its maximal degenerations. These degenerations correspond to the largest matroids lying below $M$ in the weak order. Our framework yields explicit and computable decompositions of circuit varieties that were previously out of reach for symbolic or numerical algebra systems. We apply our strategy to several classical configurations, including the V\'amos matroid, the unique Steiner quadruple system $S(3,4,8)$, projective and affine planes, the dual of the Fano matroid, and the dual of the graphic matroid of $K_{3,3}$. In each case, we successfully compute the minimal irreducible decomposition of their circuit varieties.