Bounding the number of bases of a matroid (Q5917662)

The possibility of bounding the number of bases of a matroid by a polynomial of the size \(k\) of the underlying set, or by a polynomial of the size of \(k\) times the number of circuits, is investigated. The latter holds for every member of a minor closed class of matroids if and only if the class does not contain the direct sum of an arbitrarily large number of length 2 circuits. The same condition is shown for the former question, too, if every member of the class is representable over a fixed finite field.