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On Matroids Representable over $GF(3)$
and other Fields


Author: Geoff Whittle
Journal: Trans. Amer. Math. Soc. 349 (1997), 579-603
MSC (1991): Primary 05B35
DOI: https://doi.org/10.1090/S0002-9947-97-01893-X
MathSciNet review: 1407504
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Abstract: 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 the rationals.


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Additional Information

Geoff Whittle
Affiliation: Department of Mathematics, Victoria University, PO Box 600 Wellington, New Zealand
Email: whittle@kauri.vuw.ac.nz

DOI: https://doi.org/10.1090/S0002-9947-97-01893-X
Received by editor(s): August 20, 1994
Article copyright: © Copyright 1997 American Mathematical Society

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