Inequivalent representations of matroids over prime fields

It is proved that for each prime field GF(p), there is an integer np such that a 4-connected matroid has at most np inequivalent representations over GF(p). We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term “k-coherence”.We obtain a variety of other results on inequivalent representations including the following curious one. For a prime power q, let R(q) denote the set of matroids representable over all fields with at least q elements. Then there are infinitely many Mersenne primes if and only if, for each prime power q, there is an integer mq such that a 3-connected member of R(q) has at most mq inequivalent GF(7)-representations.The theorems on inequivalent representations of matroids are consequences of structural results that do not rely on representability. The bulk of this paper is devoted to proving such results.