Combinatorial Derived Matroids

Let M be an arbitrary matroid with circuits C(M). We propose a definition of a derived matroid δM that has as its ground set C(M). Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of δM is bounded from above by |M| −r(M) and that it is connected if and only if M is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid M(K₄). We formulate conjectures relating our construction to previous definitions of derived matroids.