There are exactly seven excluded minors for the class of GF(4)--representable matroids. 1 Introduction We prove the following theorem. Theorem 1.1 A matroid M is GF(4)--representable if and only if M has no minor isomorphic to any of U 2;6 , U 4;6 , P 6 , F \Gamma 7 , F \Gamma 7 , P 8 , and P 00 8 . The definitions for these matroids, with a summary of their interesting properties, can be found in the Appendix. Other than P 00 8 , they were all known to be excluded minors for GF(4)-- representability (see Oxley [13,15]). The matroid P 00 8 is obtained by relaxing the unique pair of disjoint circuit--hyperplanes of P 8 . Ever since Whitney's introductory paper [24] on matroid theory, researchers have sought ways to distinguish the representable matroids. For any field F, the class of F--representable matroids is closed under taking minors. Thus, it is natural to characterize the minor--minimal matroids that are not F--representable; we refer to such matroids as excluded ...

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

\Lambda This paper is dedicated to Don Row who introduced all three authors to matroids. 1

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This paper surveys recent work that is aimed at generalising the results and techniques of the Graph Minors Project of Robertson and Seymour to matroids.

For a fixed finite field F and an integer k there are a finite number of matroids of branch-width k that are excluded minors for F-representability.

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M 0 of M by an element x 0 such that fx; x 0 g is independent and M 0 is unaltered by swapping the labels on x and x 0 . When x is xed, a representation of Mnx extends in at most one way to a representation of M . A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M , of the number of inequivalent F-representations of N . It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r 4, there are unique and easily described rank-r quaternary and quinternary matroids, the rst being the free spike. Finally, Seymour's Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N , whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and coxed elements occur in the initial segment of the sequence.

A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a; b 2 P, a + b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the class of matroids representable over a partial field. The matroids representable over a partial field are closed under standard matroid operations such as the taking of minors, duals, direct sums and 2-sums. Homomorphisms of partial fields are defined. It is shown that if ' : P 1! P 2 is a non-trivial partial field homomorphism, then every matroid representable over P 1 is representable over P 2. The connection with Dowling group geometries is examined. It is shown that if G is a nite abelian group, and r> 2, then there exists a partial field over which the rank{r Dowling group geometry is representable if and only if G has at most one element of order 2, that is, if G is a group in which the identity has at most two square roots.

Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Whitney’s definition embraces a surprising diversity of combinatorial structures. Moreover, matroids arise naturally in combinatorial optimization since they are precisely the structures for which the greedy algorithm works. This survey paper introduces matroid theory, presents some of the main theorems in the subject, and identifies some of the major problems of current research interest.

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