Discrete concavity and the half-plane property

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.

14 pages. The proof of Theorem 4 is corrected.

References in corpus (4)

• Polynomials with the half-plane property and matroid theory
• The Lax conjecture is true
• The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability
• On the half-plane property and the Tutte group of a matroid