Matroids Representable Over Fields With a Common Subfield

On Matroids Representable over Gf (3) and Other Fields

The matroids that are representable over GF (3) and some other fields depend on the choice of field. This paper gives matrix characterisations of the classes that arise. These characterisations are analogues of the characterisation of regular matroids as the ones that can be represented over the rationals by a totally-unimodular matrix. Some consequences of the theory are as follows. A matroid ...

Partial Fields and Matroid Representation

A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a, b ∈ P, a + b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the class of matroids representable over a partial field. The matroids representabl...

An excluded minors method for infinite matroids

The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are representable over k. We expect that the method we use to prove this will make it possible to lift m...

Counting Bases of Representable Matroids

We show that it is #P-complete to count the number of bases of matroids representable over a fixed infinite field or fields of fixed characteristic.

On matroids representable over GF(3) and other fields