Related papers: Cyclic flats of a polymatroid

Matrix representations are a powerful tool for designing efficient algorithms for combinatorial optimization problems such as matching, and linear matroid intersection and parity. In this paper, we initiate the study of matrix…

We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional root system $H_4$. This root system coincides with the vertices of the 600-cell, a 4-dimensional regular solid. We determine the automorphism group of this…

The positive existential theories of the sets $M_n(\mathbb N)$ without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorithmically undecidable. In further sections some easier…

Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of…

We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps…

We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of…

Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the…

The class of cographs or complement-reducible graphs is the class of graphs that can be generated from $K_1$ using the operations of disjoint union and complementation. By analogy, this paper introduces the class of binary comatroids as the…

In 1999 Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article we show that the conjecture generalized to matroids holds for the large class of all split matroids by…

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R=$\mathbb{Z}$, and when R is a DVR, we get…

Rook matroids were recently introduced by the author and Alexandersson as matroids whose bases arise from certain restricted rook placements on a skew-shaped board. They were shown to be a subclass of transversal matroids and positroids. We…

The Tic-Tac-Toe matroid is a paving matroid of rank $5$ on 9 elements which is pseudomodular and whose dual is non-algebraic. It has been proposed as a possible example of an algebraic matroid whose dual is not algebraic. We present an…

A topological mating is a map defined by gluing together the filled Julia sets of two quadratic polynomials. The identifications are visualized and understood by pinching ray-equivalence classes of the formal mating. For postcritically…

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…

The h-vector of a matroid M is an important invariant related to the independence complex of M and can also be recovered from an evaluation of its Tutte polynomial. A well-known conjecture of Stanley posits that the h-vector of a matroid is…

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank…

The aim of this book is to show that the use of f-analytic families of finite type cycles (cycles having finitely many irreducible components, but not compact in general) in a given complex space may be useful in complex geometry, despite…

We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial $f$ of degree $d$, there are exactly $2^{d-1}$ distinct degree $d$ polynomials with the same set of cyclic resultants as $f$.…

We study a particular graded ring structure on the set of all loopfree matroids on a fixed labeled ground set, which occurs naturally in tropical geometry. The product is given by matroid intersection and the additive structure is defined…

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…