On Inequivalent Representations of Matroids over Finite Fields

Kahn conjectured in 1988 that, for each prime powerq, there is an integern(q) such that no 3-connectedGF(q)-representable matroid has more thann(q) inequivalentGF(q)-representations. At the time, this conjecture was known to be true forq=2 andq=3, and Kahn had just proved it forq=4. In this paper, we prove the conjecture forq=5, showing that 6 is a sharp value forn(5). Moreover, we also show that the conjecture is false for all larger values ofq.