Field extensions, Derivations, and Matroids over Skew Hyperfields

Abstract

We show that a field extension $K\subseteq L$ in positive characteristic $p$ and elements $x_e\in L$ for $e\in E$ gives rise to a matroid $M^\sigma$ on ground set $E$ with coefficients in a certain skew hyperfield $L^\sigma$. This skew hyperfield $L^\sigma$ is defined in terms of $L$ and its Frobenius action $\sigma:x\mapsto x^p$. The matroid underlying $M^\sigma$ describes the algebraic dependencies over $K$ among the $x_e\in L$ , and $M^\sigma$ itself comprises, for each $m\in \mathbb{Z}^E$, the space of $K$-derivations of $K\left(x_e^{p^{m_e}}: e\in E\right)$. The theory of matroid representation over hyperfields was developed by Baker and Bowler for commutative hyperfields. We partially extend their theory to skew hyperfields. To prove the duality theorems we need, we use a new axiom scheme in terms of quasi-Pl\"ucker coordinates.