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Inequivalent representations of matroids over prime fields. (English) Zbl 1281.05041
Summary: It is proved that for each prime field $$\mathrm{GF}(p)$$, there is an integer $$n_p$$ such that a 4-connected matroid has at most $$n_p$$ inequivalent representations over $$\mathrm{GF}(p)$$. 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 $$\mathcal{R}(q)$$ 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 $$m_q$$ such that a 3-connected member of $$\mathcal{R}(q)$$ has at most $$m_q$$ inequivalent $$\mathrm{GF}(7)$$-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.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices
##### Keywords:
matroid; inequivalent representation; prime field
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##### References:
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