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 quasiPlücker coordinates.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.02447
 Bibcode:
 2018arXiv180202447P
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry;
 05B35;
 12F99;
 14T99
 EPrint:
 Changed the signing convention for coordinates to better conform to existing concepts in the literature (Tutte group, quasideterminants)