Principal Matroid Determinants

Abstract

We develop a theory of principal determinants and hypergeometric systems for realizable matroids. Our framework parallels the toric theory of Gel'fand, Kapranov, and Zelevinsky (GKZ), but with the combinatorics of matroids and their flats replacing the usual role of polytopes and their faces. In this analogy, the toric variety is replaced by a reciprocal linear space. The {principal $A$-determinant} is replaced by the {principal matroid determinant}, defined as a specialization of a resultant. The GKZ hypergeometric system is replaced by the {matroid hypergeometric system}, a holonomic $D$-module of combinatorial nature whose singular locus is conjectured to be the principal matroid~determinant.