Unlabeled equivalence for matroids representable over finite fields

We present a new type of equivalence for representable matroids that uses the automorphisms of the underlying matroid. Two $r\times n$ matrices $A$ and $A'$ representing the same matroid $M$ over a field $F$ are {\it geometrically equivalent representations} of $M$ if one can be obtained from the other by elementary row operations, column scaling, and column permutations. Using geometric equivalence, we give a method for exhaustively generating non-isomorphic matroids representable over a finite field $GF(q)$, where $q$ is a power of a prime

On the unique representability of spikes over prime fields

For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $k\in\{1, 2, ..., n - 1\}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $n \ge 2p-1$. Moreover, $M$ is uniquely representable over $GF(p)$.Comment: 8 page

On inequivalent representations of matroids over non-prime fields

For each finite field $F$ of prime order there is a constant $c$ such that every 4-connected matroid has at most $c$ inequivalent representations over $F$. We had hoped that this would extend to all finite fields, however, it was not to be. The $(m,n)$-mace is the matroid obtained by adding a point freely to $M(K_{m,n})$. For all $n \geq 3$, the $(3,n)$-mace is 4-connected and has at least $2n$ representations over any field $F$ of non-prime order $q \geq 9$. More generally, for $n \geq m$, the $(m,n)$-mace is vertically $(m+1)$-connected and has at least $2n$ inequivalent representations over any finite field of non-prime order $q\geq m^m$

On inequivalent representations of matroids over finite fields

Kahn conjectured in 1988 that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable matroid has more than n(q) inequivalent GF(q)-representations. At the time, this conjecture was known to be true for q = 2 and q = 3, and Kahn had just proved it for q = 4. In this paper, we prove the conjecture for q = 5, showing that 6 is a sharp value for n(5). Moreover, we also show that the conjecture is false for all larger values of q. © 1996 Academic Press, Inc

Inequivalent representations of ternary matroids

AbstractThis paper considers representations of ternary matroids over fields other than GF(3). It is shown that a 3-connected ternary matroid representable over a finite field F has at most ¦F¦ - 2 inequivalent representations over F. This resolves a special case of a conjecture of Kahn in the affirmative

Totally free expansions of matroids

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′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x 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 first being the free spike. Finally, Seymour\u27s 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 cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science

Fork-decompositions of matroids

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