Related papers: Excluding a line from positroids

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic…

We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high…

This paper is continuation of the paper "Primitive roots in quadratic field". We consider an analogue of Artin's primitive root conjecture for algebraic numbers which is not a unit in real quadratic fields. Given such an algebraic number,…

Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered…

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground…

The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of…

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length…

A set $S\subseteq 2^E$ of subsets of a finite set $E$ is \emph{powerful} if, for all $X\subseteq E$, the number of subsets of $X$ in $S$ is a power of 2. Each powerful set is associated with a non-negative integer valued function, which we…

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected…

Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.

We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this…

Given nonnegative integers $n$ and $d$, where $n \gg d$, what is the minimum number $r$ such that there exist linear forms $\ell_1, \ldots, \ell_r \in \mathbb{C}[x_1, \ldots, x_n]$ so that $\ell_1^d + \cdots + \ell_r^d$ is supported exactly…

We show that, for each real number $\alpha > 0$ and odd integer $k\ge 5$ there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M| \ge \alpha 2^{r(M)}$ and with no $k$-element circuit, then $M$ has critical number at…

It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1,2,...,n without repetition. These labellings are called S_n EL-labellings,…

The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not.…

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as…

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…

A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

For a positive integer $N$, we define the N-rank of a non singular integer $d\times d$ matrix $A$ to be the maximum integer $r$ such that there exists a minor of order $r$ whose determinant is not divisible by $N$. Given a positive integer…