Related papers: Perfect matroids over hyperfields

A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and…

Let $G$ be a linear algebraic group over an infinite field $k$. Loosely speaking, a $G$-torsor over $k$-variety is said to be versal if it specializes to every $G$-torsor over any $k$-field. The existence of versal torsors is well-known. We…

We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant,…

Let $X$ be an $(8k+i)$-dimensional pathwise connected $CW$-complex with $i=1$ or $2$ and $k\ge0$, $\xi$ be a real vector bundle over $X$. Suppose that $\xi$ admits a stable complex structure over the $8k$-skeleton of $X$. Then we get that…

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook…

Stacks have become a prevalent tool in studying problems with connections to String Theory, hence we see a need to develop a theory of supersymmetric stacks proper. We first define derived stacks on $\mathbb{Z}_2$-bi-graded k-modules…

Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the…

Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally attaches an ideal of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of…

Rough sets are efficient for data pre-processing in data mining. As a generalization of the linear independence in vector spaces, matroids provide well-established platforms for greedy algorithms. In this paper, we apply rough sets to…

This article presents a simple characterization for entangled vectors in a finite dimensional Hilbert space $H$. The characterization is in terms of the coefficients of an expansion of the vector relative to an orthonormal basis for $H$.…

This paper studies the structure of finite hyperfields $H$, and finds a subtle pattern in their addition operation. Consider the class $\mathcal{H}$ of all hyperfields with a given multiplicative group on $H^\times = H - \{0\}$ and given…

Generalizing a theorem of the first two authors and Geelen for planes, we show that, for a real-representable matroid $M$, either the average hyperplane-size in $M$ is at most a constant depending only on its rank, or each hyperplane of $M$…

We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We exhibit a…

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…

Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a r-dimensional…

We study the behavior of $h$-vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the $h$-vector of the order complex of the lattice of flats of a matroid is component-wise…

One of the most important classes of even $\Delta$-matroids arises from orientable ribbon graphs, which play a role analogous to that of graphic matroids in matroid theory. Motivated by a natural correspondence between strong…

This paper investigates infinite matroids from a categorical perspective. We prove that the category of infinite matroids is a proto-exact category in the sense of Dyckerhoff and Kapranov, thereby generalizing our previous result on the…

For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of…

We show that if the ground set of a matroid can be partitioned into $k\ge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may…