Stability, fragility, and Rota's Conjecture

TL;DR

This paper explores the relationships between Rota's conjecture, matroid fragility, and the Bounded Canopy Conjecture, establishing implications for finite excluded minors and the stabilization of matroid classes over finite fields.

Contribution

It proves that a false Rota's conjecture implies either the Bounded Canopy Conjecture is false or an infinite chain of non-stabilized matroids exists, linking these conjectures and extending known results.

Findings

Rota's conjecture holds for GF(4).

Classes of near-regular and sixth-roots-of-unity have finitely many excluded minors.

For GF(5), Rota's conjecture reduces to the Bounded Canopy Conjecture.

Abstract

Fix a matroid N. A matroid M is N-fragile if, for each element e of M, at least one of M\e and M/e has no N-minor. The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N-minor and are N-fragile have branch width bounded by a constant depending only on q and N. A matroid N stabilizes a class of matroids over a field F if, for every matroid M in the class with an N-minor, every F-representation of N extends to at most one F-representation of M. We prove that, if Rota's conjecture is false for GF(q), then either the Bounded Canopy Conjecture is false for GF(q) or there is an infinite chain of GF(q)-representable matroids, each not stabilized by the previous, each of which can be extended to an excluded minor. Our result implies the previously known result that Rota's conjecture holds for GF(4), and that the classes of near-regular and sixth-roots-of-unity have a finite number of excluded minors. However, the bound that we obtain on the size of such excluded minors is considerably larger than that obtained in previous proofs. For GF(5) we show that Rota's Conjecture reduces to the Bounded Canopy Conjecture.