On matroid representability and minor problems

Abstract

In this paper we look at complexity aspects of the following problem (matroid representability) which seems to play an important role in structural matroid theory: Given a rational matrix representing the matroid M, the question is whether M can be represented also over another specific finite field. We prove this problem is hard, and so is the related problem of minor testing in rational matroids. The results hold even if we restrict to matroids of branch-width three.