相关论文: Matroid Products in Tropical Geometry
In this paper, we propose a uniform approach to tackle problems about squarefree monomial ideals whose powers have good properties. We employ this approach to achieve a twofold goal: (i) recover and extend several well--known results in the…
We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric…
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…
This paper investigates entropic matroids, that is, matroids whose rank function is given as the Shannon entropy of random variables. In particular, we consider $p$-entropic matroids, for which the random variables each have support of…
Results of Macdonald and Dold from the 1960s and '70s expressing the Lefschetz numbers of symmetric powers of a self-map of a compact ENR in terms of the Lefschetz numbers of iterates of the map are extended using the notion of a…
We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying…
Recently, the relationship between matroids and generalized rough sets based on relations has been studied from the viewpoint of linear independence of matrices. In this paper, we reveal more relationships by the predecessor and successor…
We introduce quivers of valuated matroids and study their tropical parameter spaces. We define quiver Dressians, which parametrize containment of tropical linear spaces after tropical matrix multiplication, and show that tropicalizations of…
We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalised symmetric groups. We adapt the methods of Savitt and Stanley from their paper `A note on the symmetric…
For a discrete poset $\mathcal X$ McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$ from the order complex to the poset equipped with the Up topology is a weak homotopy equivalence. Much later, Zivaljevi\'{c} defined the…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of…
We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$…
A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms…
We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry. We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set.…
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…
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…
The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of…
A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lov\'asz initiated the study of matroids from…
In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More…