Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra
SIAM Journal on Discrete Mathematics
Discrete Convex Analysis: Monographs on Discrete Mathematics and Applications 10
Discrete Convex Analysis: Monographs on Discrete Mathematics and Applications 10
M-Convex Functions on Jump Systems: A General Framework for Minsquare Graph Factor Problem
SIAM Journal on Discrete Mathematics
A Steepest Descent Algorithm for M-Convex Functions on Jump Systems
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Operations on M-Convex Functions on Jump Systems
SIAM Journal on Discrete Mathematics
On the half-plane property and the Tutte group of a matroid
Journal of Combinatorial Theory Series B
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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 nonvanishing 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.