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Excluded-minor structure for κ-bounded matroids

Identify additional excluded minors for the class of matroids representable by real matrices whose circuit imbalance measure κ(A) is bounded, with the aim of improving upper bounds on the number of elements of κ-bounded matroids beyond those implied by excluding a U_{2,l} (line) minor.

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Background

The authors use minor-closedness and exclusion of long lines (U_{2,l}) to obtain polynomial bounds on the number of elements in κ-bounded matroids, paralleling techniques developed for Δ-modular matroids.

They note that several excluded-minor results are known for Δ-modular matrices, but some do not carry over directly to κ-bounded matroids due to closure under direct summation; κ is also self-dual, suggesting that duality may be leveraged. Discovering further excluded minors specific to κ-bounded matroids could tighten current column bounds.

References

In terms of open questions, many still remain. One natural question is whether we can find other excluded minors in terms of the imbalance measure kappa_A in order to get closer to the right bound on the number of elements of a kappa-bounded matroid.

— Excluding a Line Minor via Design Matrices and Column Number Bounds for the Circuit Imbalance Measure (2510.20301 - Dadush et al., 23 Oct 2025) in Subsection: Open Questions