Discrete concavity and the half-plane property

Details Discrete concavity and the half-plane property Discrete concavity and the half-plane property

2009-04-02

arxiv.org/abs/0904.0363v3

SIAM J. Discrete Math. 24 (2010), no. 3, 921-933

Mathematics

Combinatorics

14 pages. The proof of Theorem 4 is corrected.

Scientific paper

10.1137/090758738

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.

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