Related papers: An excluded minors method for infinite matroids
In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This…
We provide a new axiom system for flag matroids, characterize representability of uniform flag matroids, and give forbidden minor characterizations of full flag matroids that are representable over $\mathbb{F}_2$ and $\mathbb{F}_3$ along…
An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other…
We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary…
A lattice path matroid is a transversal matroid for which some collection of incomparable intervals in some linear order on the ground set is a presentation. We characterize the minor-closed class of lattice path matroids by its excluded…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…
Let $\Lambda$ be a finite-dimensional algebra over an algebraically closed field, then $\Lambda$ is either tame or wild. Is there any homological description in terms of AR-translations on tameness? Or equivalently, is there any…
A result of Seymour implies that any 3-connected matroid with a modular 3-point line is binary. We prove a similar characterization for 3-connected matroids with modular 4-point lines. We show that such a matroid is either representable…
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…
We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. The proof is based on the technique of matrix problems (boxes and reduction algorithm). It implies, in particular,…
A super-minimally $k$-connected matroid is a $k$-connected matroid having no proper $k$-connected restriction of size at least $2k-2$. This extends the corresponding concept for graphs. For $k=2$ and $k=3$, we determine the maximum size of…
We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…
A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is…
We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in monadic…
We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The…
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle,…
We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for {\em tame} vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the…
We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…
In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove…
A matroid of rank $r$ on $n$ elements is a positroid if it has a representation by an $r$ by $n$ matrix over $\mathbb{R}$, each $r$ by $r$ submatrix of which has nonnegative determinant. Earlier characterizations of connected positroids and…