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 is representable over GF (3) and GF (5) if and only if it is representable over GF (3) and the rationals, and this holds if and only if it is representable over GF (p) for all odd primes p. A matroid is representable over GF (3) and the complex numbers if and only if it is representable over GF (3) and GF (7). A matroid is representable over GF (3), GF (4) and GF (5) if and only if it is representable over every field except possibly GF (2). If a matroid is representable over GF (p) for all odd primes p, then it is representable over

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