DISCRETE CONCAVITY AND THE HALF-PLANE PROPERTY.

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 well 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. [ABSTRACT FROM AUTHOR]