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

We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory…

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that $s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In…

This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…

We show algorithms for computing representative families for matroid intersections and use them in fixed-parameter algorithms for set packing, set covering, and facility location problems with multiple matroid constraints. We complement our…

For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank…

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with…

A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…

We show that, for any prime $p$ and integer $k \geq 2$, a simple GF($p$)-representable matroid with sufficiently high rank has a rank-$k$ flat which is either independent in $M$, or is a projective or affine geometry. As a corollary we…

We study smoothness of realization spaces of matroids for small rank and ground set. For $\mathbb{C}$-realizable matroids, when the rank is $3$, we prove that the realization spaces are all smooth when the ground set has $11$ or fewer…

Let $M=(E,{\cal I})$ be a matroid, and let $\cal S$ be a family of subsets of size $p$ of $E$. A subfamily $\widehat{\cal S}\subseteq{\cal S}$ represents ${\cal S}$ if for every pair of sets $X\in{\cal S}$ and $Y\subseteq E\setminus X$ such…

It is proved that, for a prime number $p$, showing that an $n$-element matroid is not representable over $GF(p)$ requires only $O(n^2)$ rank evaluations.

Let M to be a matroid defined on a finite set E. A subset L of E is locked in M if L is 2-connected in M, the complement of L is 2-connected in the dual M*, and min{r(L), r*(complement of L)} is greater than 1. In this paper, we prove that…

The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Pl\"ucker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a…

Let K be a field, let R=K[x_1,..., x_m] be a polynomial ring with the standard Z^m-grading (multigrading), let L be a Noetherian multigraded R-module, and let F: E --> G be a finite free multigraded presentation of L over R. Given a choice…

One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures…

Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as…

A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engstr\"om recently showed how to…

Given a group $G$ of automorphisms of a matroid $M$, we describe the representations of $G$ on the homology of the independence complex of the dual matroid $M^*$. These representations are related with the homology of the lattice of flats…

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid,…