The Chow form of a reciprocal linear space

TL;DR

This paper provides a determinantal representation of the Chow form for reciprocal linear spaces, linking algebraic geometry, matroid theory, and hyperbolic varieties with applications to optimization and combinatorics.

Contribution

It introduces a definite determinantal formula for the Chow form of reciprocal linear spaces, revealing new geometric and algebraic properties and connections to hyperbolic varieties and the positive Grassmannian.

Findings

Determinantal representation of the Chow form established

Existence of symmetric rank-one Ulrich sheaves shown

Representation of the entropic discriminant as a sum of squares

Abstract

A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.