Inequivalent Representations of Bias Matroids

Suppose that q is a prime power exceeding five. For every integer N there exists a 3-connected GF(q)-representable matroid, in particular, a free spike or a free swirl, that has at least N inequivalent GF(q)-representations. In contrast to this, Geelen, Oxley, Vertigan and Whittle have conjectured that, for any integer r > 2, there exists an integer n(q, r) such that if M is a 3-connected GF(q)-representable matroid and M has no rank-r free-swirl or rank-r free-spike minor, then M has at most n(q, r) inequivalent GF(q)-representations. The main result of this paper is a proof of this conjecture for Zaslavsky’s class of bias matroids.