The Entropic Discriminant

TL;DR

This paper explores the geometric and algebraic properties of the entropic discriminant, a polynomial linked to matrices, revealing its degree, singularities, and sum of squares representation, with applications across various mathematical fields.

Contribution

It provides a detailed geometric analysis of the entropic discriminant and expresses its degree using the characteristic polynomial of the associated matroid, including a sum of squares form in the corank-one case.

Findings

Degree of the entropic discriminant expressed via matroid characteristic polynomial

Singularities of reciprocal linear spaces are crucial to understanding its geometry

Sum of squares representation derived for the corank-one case

Abstract

The entropic discriminant is a non-negative polynomial associated to a matrix. It arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry. It describes the complex branch locus of the polar map of a real hyperplane arrangement, and it vanishes when the equations defining the analytic center of a linear program have a complex double root. We study the geometry of the entropic discriminant, and we express its degree in terms of the characteristic polynomial of the underlying matroid. Singularities of reciprocal linear spaces play a key role. In the corank-one case, the entropic discriminant admits a sum of squares representation derived from the discriminant of a characteristic polynomial of a symmetric matrix.