Principal Matroid Determinants

TL;DR

This paper introduces a new theoretical framework for principal determinants and hypergeometric systems based on realizable matroids, paralleling GKZ toric theory but with matroid combinatorics.

Contribution

It develops the concept of principal matroid determinants and matroid hypergeometric systems, extending toric theory to matroid combinatorics and reciprocal linear spaces.

Findings

Defined principal matroid determinants as specializations of resultants.

Introduced matroid hypergeometric systems as holonomic D-modules.

Conjectured the singular locus of the system is the principal matroid determinant.

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.