On Matroids Representable over Gf (3) and Other Fields

On matroids representable over GF(3) and other fields

Matroids Representable Over Fields With a Common Subfield

A matroid is GF(q)-regular if it is representable over all proper superfields of the field GF(q). We show that, for highly connected matroids having a large projective geometry over GF(q) as a minor, the property of GF(q)-regularity is equivalent to representability over both GF(q) and GF(q) for some odd integer t ≥ 3. We do this by means of an exact structural description of

Orienting Matroids Representable Over Both GF(3) and GF(5)

For matroids representable over both GF 3] and GF 5], we provide a recipe for constructing an orientation.

Lifts of matroid representations over partial fields

There exist several theorems which state that when a matroid is representable over distinct fields F1, . . . ,Fk, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle’s characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a m...

On the Uniqueness of Matroid Representations over Gf(4)

We determine when a matroid is uniquely representable over GF(4). In particular all 3-connected (GF(4)-representable) matroids have this property.