Related papers: Tree metrics and log-concavity for matroids
The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $ and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus X_i)…
The aim of this paper is to develop a $(q,m)$-polymatroidal approach to higher supports and higher rank-weight enumerators of rank-metric codes. In this framework, we establish analogs of several fundamental results known for matroids and…
We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter…
We investigate possible linear, algebraic, and Frobenius flock characteristic sets of matroids. In particular, we classify possible combinations of linear and algebraic characteristic sets when the algebraic characteristic set is finite or…
We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily 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…
The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials,…
We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of…
Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link…
We study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan-Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module,…
We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group $(G,+)$ is a bijection $f:A\to B$ between two finite subsets $A,B$ of $G$ satisfying $a+f(a)\notin A$ for all $a\in A$. A…
We introduce the Hermitian-invariant group $\Gamma_f$ of a proper rational map $f$ between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define…
Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for…
The tree-width of a multivariate polynomial is the tree-width of the hypergraph with hyperedges corresponding to its terms. Multivariate polynomials of bounded tree-width have been studied by Makowsky and Meer as a new sparsity condition…
Connections between $q$-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel's $q$-hit polynomial. Both this new statistic $mat$ and another statistic for the $q$-hit polynomial…
We prove that the existence of an automorphism of finite order on a (defined over a number field) variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special…
We introduce $q$-analogues of uniform matroids, which we call $q$-niform matroids. While uniform matroids admit actions of symmetric groups, $q$-niform matroids admit actions of finite general linear groups. We show that the equivariant…
We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and…
To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique…
We extend the results of Zhang et al. to show that $\lambda$ is an eigenvalue of a $k$-uniform hypertree $(k \geq 3)$ if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to…