Stability, fragility, and Rotaʼs Conjecture

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-ofunity 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. ∗Parts of this paper were previously published in the third author’s PhD thesis [33]. The research of all authors was partially supported by a grant from the Marsden Fund of New Zealand. The first author was also supported by a FRST Science & Technology post-doctoral fellowship. The third author was also supported by the Netherlands Organisation for Scientific Research (NWO). †School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand. E-mail: [email protected], Geoff.Whittle@msor. vuw.ac.nz ‡Centrum Wiskunde en Informatica, Postbus 94079, 1090 GB Amsterdam, The Netherlands. E-mail: [email protected]