Discrete concavity and the half-plane property

TL;DR

This paper explores the connection between discrete concavity, tropical geometry, and polynomials with the half-plane property, revealing new insights into M-concave functions and providing a proof of Speyer's hive theorem.

Contribution

It introduces a family of M-concave functions derived from polynomials with the half-plane property and analyzes their relationship using tropical geometry, extending the theory of discrete convex analysis.

Findings

The space of polynomials with the half-plane property is strictly contained in the space of M-concave functions.

A short proof of Speyer's hive theorem is provided.

The study links tropical geometry with discrete convex analysis and polynomial non-vanishing properties.

Abstract

Murota et al. have recently developed a theory of discrete convex analysis which concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over a field of generalized Puiseux series) with prescribed non-vanishing properties. This family contains several of the most studied M-concave functions in the literature. In the language of tropical geometry we study the tropicalization of the space of polynomials with the half-plane property, and show that it is strictly contained in the space of M-concave functions. We also provide a short proof of Speyer's hive theorem which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices.