Discrete Concavity and the Half-Plane Property

Murota et al. have recently developed a theory of discrete convex analysis which concerns M -convex and L-convex functions on jump systems. We introduce here a family of M -concave functions arising naturally from polynomials (over the field of Puiseux series) with prescribed non-vanishing properties. This family contains several of the most studied M -concave functions in the literature. 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.