Related papers: Girth in $GF(q)$-representable matroids

We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power $q$ and constant $k$, the characteristic polynomial of any loopless, $GF(q)$-representable matroid with…

The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an integer $n \ge 0$ is defined to be the maximum number of elements in a simple matroid…

Let ${\mathbb{F}_{q}}$ be the finite field of order $q$. Let $G$ be one of the three groups ${\rm GL}(n, \mathbb{F}_q)$, ${\rm SL}(n, \mathbb{F}_q)$ or ${\rm U}(n, \mathbb{F}_q)$ and let $W$ be the standard $n$-dimensional representation of…

We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles. This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to…

There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable…

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for…

We show for each positive integer $a$ that, if $\cM$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in $\cM$ can be covered by…

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear…

For a finite multigraph G, the reliability function of G is the probability R_G(q) that if each edge of G is deleted independantly with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a…

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number…

We prove that there is no polynomial $p(\cdot)$ with the property that a matroid $M$ can be determined to be either a lifted-graphic or frame matroid using at most $p(|M|)$ rank evaluations. This resolves two conjectures of Geelen, Gerards…

We consider matroids with the property that every subset of the ground set of size $s$ is contained in a $2s$-element circuit and every subset of size $t$ is contained in a $2t$-element cocircuit. We say that such a matroid has the…

We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Q\cup R\cup S\cup T)$ such that there are two ways to reduce $G$ by a…

Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that…

We construct, for every $r \ge 3$ and every prime power $q > 10$, a rank-$r$ matroid with no $U_{2,q+2}$-minor, having more hyperplanes than the rank-$r$ projective geometry over $\mathrm{GF}(q)$.

We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let $\mathbb F$ be a finite field. The first conjecture states that: the branch-width of any $\mathbb F$-representable $N$-fragile matroid is…

Let G be a finite group and V a finite-dimensional rational G-representation. We ask whether there exists a finite Galois extension L/K of number fields with Galois group G, an elliptic curve E/K, and a G-submodule of E(L) tensor Q…

Let $\K$ be a finite extension of a characteristic zero field $\F$. We say that the pair of $n\times n$ matrices $(A,B)$ over $\F$ represents $\K$ if $\K \cong \F[A]/< B >$ where $\F[A]$ denotes the smallest subalgebra of $M_n(\F)$…

We consider the GF$(4)$-representable matroids with a circuit-hyperplane such that the matroid obtained by relaxing the circuit-hyperplane is also GF$(4)$-representable. We characterize the structure of these matroids as an application of…

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the…