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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