## A Characterisation of the Matroids Representable over GF(3) and the Rationals

#### Abstract

It follows from a fundamental (1958) result of Tutte that a binary
matroid is representable over
the rationals if and only if it can be represented by a unimodular matrix, that
is, by a matrix over the rationals with the property that all subdeterminants
belong to {0,1,-1}. For an arbitrary field **F** it is of interest to ask for
a matrix characterisation of those matroids representable over **F** and the
rationals. In this paper this question is answered when **F** is **GF(3)**. It is
shown that a ternary matroid is representable over the rationals if and only
if it can be represented over the rationals by a matrix **A** with the property
that all subdeterminants of **A** belong to the set {0, 2^i: i an integer}
. While ternary matroids are uniquely representable over **GF(3)**, this is not
generally the case for representations of ternary matroids over other fields.
A characterisation is given of the class of 3--connected ternary matroids that
are not uniquely representable over the rationals.
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Postscript version