14 problems

Let $(G,\mathcal{B})$ be a biased graph, let $M$ be an elementary lift of the frame matroid $F(G,\mathcal{B})$, and suppose that $M$ is $3$-connected. Let $\mathbb{F}$ be a field a…

Let $\mathbb{F}$ be a field, let $G$ be a loopless graph with orientation $D$, and let $\psi$ and $\psi'$ be $(\mathbb{F}^+\rtimes \mathbb{F}^{\times})$-gain functions for $G$. Sup…

Let $\mathbb{P}=(R,G)$ be a skew partial field, let $n,r,s$ be positive integers with $s\geq r$, define … and let $A$ be an $X\times E$ weak $\mathbb{P}$-matrix. A map…

Let $M$ be a 2-uniform matroid, given by a representation over the 2-uniform partial field. Basis-counting conjecture. There is a polynomial-time algorithm to count the number of b…

A 2-uniform matroid is a matroid in the class of matroids represented over the 2-uniform partial field, and a QU matroid is a quaternionic unimodular matroid. 2-uniform matroid con…

Let $\mathbb{H}$ be the quaternions, with norm $|\cdot|$, and let $p,q,r\in\mathbb{H}$ satisfy … for all distinct $i,j\in\{0,1,p,q,r\}$. Let $M$ be a quaternionic unimodular matroi…

Let $\mathbb{P}$ be a skew partial field, and let $M_1$ and $M_2$ be $\mathbb{P}$-representable matroids having a common flat $N$ and representations that agree on $N$. If $N$ is a…

Let $M$ be a matroid representable over a skew partial field, and let a quadruple of subsets of $M$ be given. Ingleton inequality conjecture. Ingleton's Inequality is satisfied by…

A skew partial field is a partial field whose multiplicative group need not be commutative. Let $\mathbb{P}(n,\mathbb{F})$ denote the partial field associated with $n\times n$ matr…

Let ${\mathbb{P}}$ be a partial field, let $M$ be a 3-connected ${\mathbb{P}}$-representable matroid, and let $N=M{-} e$ be a 3-connected deletion that settles $M$. Let $I_{N,M}$ b…

Let $M$ be a representable matroid and let $e$ be an element of $M$. The deletion $M{-} e$ settles $M$ when representations of $M{-} e$ determine representations of $M$ in the rele…

Let ${\mathbb{P}}$ be a partial field, and let ${\cal M}$ be a class of matroids. Say that ${\mathbb{P}}$ is ${\cal M}$-universal if, for every partial field ${\mathbb{P}}'$ such t…

Let $N$ be a 3-connected matroid. A matroid is ${\mathbb{P}}_N$-representable when it has a representation over the universal partial field ${\mathbb{P}}_N$. The matroid $N$ is a…

Let $N$ be a matroid and let ${\mathbb{P}}_N$ denote its universal partial field. A ${\mathbb{P}}_N$-representation of $N$ is a representation over ${\mathbb{P}}_N$, and two such r…