Related papers: On the Ehrhart Polynomial of Minimal Matroids

We generalize R. P. Stanley's celebrated theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to…

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$.…

Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…

For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the…

In a finite real reflection group, two factorizations of a Coxeter element into an arbitrary number of reflections are shown to lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes.…

In this paper we prove that the Stanley--Reisner ideal or cover ideal $I$ of a matroid is minimally resolvable by iterated mapping cones. As a technical tool for this purpose, we introduce and study focal matroids, which are submatroids of…

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^*$-polynomial both succeed and fail to be…

We give explicit recursive constructions for the polytope of all matroids $\Omega_{r,n}$ in ranks 2 and 3 for all ground set sizes. This polytope was introduced in recent work by Ferroni and Fink as a tool for checking positivity…

In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-matroid and considering the projective space as…

We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$…

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave: 1. The matroid complex of independent subsets of E. 2. The broken circuit complex of…

We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the…

Let $P$ be a polytope. The hitting number of $P$ is the smallest size of a hitting set of the facets of $P$, i.e., a subset of vertices of $P$ such that every facet of $P$ has a vertex in the subset. An extended formulation of $P$ is the…

The problem of finding the minimum rank of a matrix with a given zero-nonzero pattern has been generalized to a class of matroids associated to the pattern. The fundamental lower bound known as the triangle number still holds in this…

A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ --- that is, a "polynomial" in which the coefficients are…

Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…

We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent…

Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations,…

We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the…

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when…