15 problems

Let $\mathbb{P}$ be a partial field, and let $\mathbb{L}\mathbb{P}$ denote its lift, constructed from the fundamental elements of $\mathbb{P}$. Pendavingh's idempotence conjecture.…

The excluded-minor conjecture for 2-regular matroids. This is the complete list of excluded minors for the class of 2-regular matroids; equivalently, there are no excluded minors f…

Let $M$ be a matroid. A matroid is representable over all fields of size at least $4$ when it has a representation over every field whose cardinality is at least $4$. Let…

Let $\mathbb{P}=(R,G)$ be a skew partial field, and let $M$ be a matroid that is $\mathbb{P}$-representable, meaning that it arises from a $\mathbb{P}$-chain group as in the source…

Let $\mathbb{P}$ be a partial field, and let $\operatorname{\mathcal{M}}(\mathbb{P})$ denote the class of $\mathbb{P}$-representable matroids. Partial-field inequivalent-representa…

A partial field $\mathbb{P}$ is finitary if it admits a homomorphism to $\operatorname{GF}(q)$ for some prime power $q$. Let $\operatorname{\mathcal{M}}(\mathbb{P})$ denote the cla…

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 ${\mathbb{P}}$ 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…

Let $\mathbb{G}$ denote the golden-ratio partial field. Golden-ratio partial-field conjecture. A matroid is representable over $\operatorname{GF}(4)\otimes\mathbb{R}$ if and only i…

Uniform partial-field characterization conjecture. A matroid is representable over $\operatorname{GF}(2^k)$ for all $k>1$ if and only if it is representable over…

Let $\mathbb{P}$ be a partial field, and let $\mathbb{L}\mathbb{P}$ denote its lift. Lift idempotence conjecture. … The conjecture is related to whether the canonical homomorphism…

$\mathbb{P}_4$ characterization conjecture. A matroid $M$ is representable over all finite fields with at least $4$ elements if and only if $M$ is representable over $\mathbb{P}_4$…

Let $k$ be a prime power. The claim concerns matroids and representability over fields and finite fields $\operatorname{GF}(q)$. Finite-field list conjecture. There exists a number…