Excluded Minors for Matroid Representability
by
Charles Semple
Victoria University of Wellington

A matroid is a collection of subsets of a finite set that satisfies certain properties. These properties are an axiomatization of the notion of independence. For example, the linearly independent subsets of a finite set of vectors from a vector space over a field is such a collection of subsets. Matroids of this type are called representable matroids.

If a matroid M can be realized as the linearly independent subsets of a finite multiset of vectors from a vector space over a field F, then M is said to be representable over F. One of the more difficult problems in matroid theory is to characterize the class of matroids representable over a particular field. A commonly sought way of doing this is via excluded minors. Rota conjectures that, for all prime powers q, the list of excluded minors for the class of matroids representable over GF(q) is finite. This talk surveys results in connection with Rota's conjecture. In particular, we present a recent result of Oxley, Semple, and Vertigan.

Date received: June 11, 1998