Rudi Pendavingh: Field extensions, derivations, and matroids over skew hyperfields

The theory of matroid representation over hyperfields was developed by Baker and Bowler for commutative hyperfields. We partially extend their theory to skew hyperfields, using a new axiom scheme in terms of quasi-Plucker coordinates.
The immediate motivation for making this generalization comes from the study of algebraic matroids. An algebraic matroid arises from a finite set E, a field extension L/K, and an element x_e in L for each e in E. Then the algebraic matroid M(K,x) has ground set E, and a subset F of E is dependent in the matroid exactly if {x_e : e in F} is algebraically dependent over K.
Previous work of Bollen, Draisma and Pendavingh showed that if K has positive characteristic, then an algebraic representation (K,x) induces a matroid valuation of M(K,x), called the Lindström valuation. It was show by Cartwright that this valuation determines the inseparable index of subfields K(x_e: e in B) in L, for each basis B of M(K,x).
We show that (K,x) in fact gives a matroid over a certain skew hyperfield, which comprises more detailed information such as the the space of K-derivations of K(x_e: e in E), as well as the Lindström valuation.