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On inequivalent representations of matroids over finite fields. (English) Zbl 0856.05021
It was conjectured by J. Kahn [On the uniqueness of matroid representations over GF(4), Bull. Lond. Math. Soc. 20, No. 1, 5-10 (1988; Zbl 0609.05028)] that, for each prime power $$q$$, there is an integer $$n(q)$$ such that no 3–connected $$\text{GF} (q)$$-representable matroid has more than $$n(q)$$ inequivalent $$\text{GF} (q)$$-representations. At the time, this conjecture was known to be true for $$q = 2$$ and $$q = 3$$, and Kahn had just proved it for $$q = 4$$. This paper proves the conjecture for $$q = 5$$ showing that 6 is a sharp value for $$n(5)$$. Moreover, it is also shown that the conjecture is false for all larger values of $$q$$.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices
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