Related papers: Representing Matroids over the Reals is $\exists \…

Generalizing a theorem of the first two authors and Geelen for planes, we show that, for a real-representable matroid $M$, either the average hyperplane-size in $M$ is at most a constant depending only on its rank, or each hyperplane of $M$…

We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily a…

We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for…

In this paper we give a necessary and sufficient criterion for representability of a matroid over an algebraic closed field. This leads to an algorithm, based on an extension of Groebner Bases, in order to decide if a given matroid is…

We study realization spaces of matroids over hyperfields (in the sense of Baker and Bowler). More precisely, given a matroid M and a hyperfield H we determine the space of all H-matroids over M. This can be seen as the matroid stratum of…

There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First,…

Let $r \leqslant n$ be nonnegative integers, and let $N = \binom{n}{r} - 1$. For a matroid $M$ of rank $r$ on the finite set $E = [n]$ and a partial field $k$ in the sense of Semple--Whittle, it is known that the following are equivalent:…

We introduce the families of solvable and nilpotent matroids, examining their realization spaces, closures, and associated matroid and circuit varieties. We study their realizability, as well as the irreducible decomposition of their…

Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given…

This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…

We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems…

A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the…

We introduce various quantities that can be defined for an arbitrary matroid, and show that certain conditions on these quantities imply that a matroid is not representable over $\mathbb{F}_q$ where $q$ is a prime power. Mostly, for a…

We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q,…

The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…

A transversal matroid $M$ of rank $r$ on $[n]$ can be associated to a family of binary matrices corresponding to different presentations of $M$. We describe those matrices which arise from unique maximal presentations of size $r$- giving a…

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if…

We introduce a procedure that solves the decision problem whether a given matroid M is a gammoid. The procedure consists of three pieces: First, we introduce a notion of a valid matroid tableau which captures the current state of knowledge…

It is proved that for each prime field $GF(p)$, there is an integer $f(p)$ such that a 4-connected matroid has at most $f(p)$ inequivalent representations over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion for…

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